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Theorem sltres 25620
Description: If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)
Assertion
Ref Expression
sltres  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( A  |`  X ) < s ( B  |`  X )  ->  A < s B ) )

Proof of Theorem sltres
Dummy variables  a  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noreson 25616 . . . . . . 7  |-  ( ( A  e.  No  /\  X  e.  On )  ->  ( A  |`  X )  e.  No )
213adant2 977 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  ( A  |`  X )  e.  No )
3 noreson 25616 . . . . . . 7  |-  ( ( B  e.  No  /\  X  e.  On )  ->  ( B  |`  X )  e.  No )
433adant1 976 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  ( B  |`  X )  e.  No )
5 sltintdifex 25619 . . . . . . 7  |-  ( ( ( A  |`  X )  e.  No  /\  ( B  |`  X )  e.  No )  ->  (
( A  |`  X ) < s ( B  |`  X )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  _V ) )
6 onintrab 4782 . . . . . . 7  |-  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  _V 
<-> 
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  On )
75, 6syl6ib 219 . . . . . 6  |-  ( ( ( A  |`  X )  e.  No  /\  ( B  |`  X )  e.  No )  ->  (
( A  |`  X ) < s ( B  |`  X )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  On ) )
82, 4, 7syl2anc 644 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( A  |`  X ) < s ( B  |`  X )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  On ) )
98imp 420 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  On )
10 simpl3 963 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  X  e.  On )
11 sltval2 25612 . . . . . . . . . . . 12  |-  ( ( ( A  |`  X )  e.  No  /\  ( B  |`  X )  e.  No )  ->  (
( A  |`  X ) < s ( B  |`  X )  <->  ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) ) )
122, 4, 11syl2anc 644 . . . . . . . . . . 11  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( A  |`  X ) < s ( B  |`  X )  <->  ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) ) )
13 fvex 5743 . . . . . . . . . . . . 13  |-  ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
14 fvex 5743 . . . . . . . . . . . . 13  |-  ( ( B  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
1513, 14brtp 25373 . . . . . . . . . . . 12  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } ) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  <-> 
( ( ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  /\  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )  \/  (
( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  \/  ( ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o ) ) )
16 1n0 6740 . . . . . . . . . . . . . . . . . 18  |-  1o  =/=  (/)
17 df-ne 2602 . . . . . . . . . . . . . . . . . 18  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
1816, 17mpbi 201 . . . . . . . . . . . . . . . . 17  |-  -.  1o  =  (/)
19 eqeq1 2443 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  ->  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  <->  1o  =  (/) ) )
2018, 19mtbiri 296 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  ->  -.  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )
21 ndmfv 5756 . . . . . . . . . . . . . . . 16  |-  ( -. 
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X )  ->  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )
2220, 21nsyl2 122 . . . . . . . . . . . . . . 15  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  ->  |^|
{ a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X ) )
2322adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X ) )
2423orcd 383 . . . . . . . . . . . . 13  |-  ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )  -> 
( |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( A  |`  X )  \/  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X ) ) )
2522adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( A  |`  X ) )
2625orcd 383 . . . . . . . . . . . . 13  |-  ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  ( A  |`  X )  \/  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( B  |`  X ) ) )
27 2on 6733 . . . . . . . . . . . . . . . . . . . . 21  |-  2o  e.  On
2827elexi 2966 . . . . . . . . . . . . . . . . . . . 20  |-  2o  e.  _V
2928prid2 3914 . . . . . . . . . . . . . . . . . . 19  |-  2o  e.  { 1o ,  2o }
3029nosgnn0i 25615 . . . . . . . . . . . . . . . . . 18  |-  (/)  =/=  2o
31 df-ne 2602 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  =/=  2o  <->  -.  (/)  =  2o )
3230, 31mpbi 201 . . . . . . . . . . . . . . . . 17  |-  -.  (/)  =  2o
33 eqeq1 2443 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o  ->  ( ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  <->  2o  =  (/) ) )
34 eqcom 2439 . . . . . . . . . . . . . . . . . 18  |-  ( 2o  =  (/)  <->  (/)  =  2o )
3533, 34syl6bb 254 . . . . . . . . . . . . . . . . 17  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o  ->  ( ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  <->  (/)  =  2o ) )
3632, 35mtbiri 296 . . . . . . . . . . . . . . . 16  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o  ->  -.  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )
37 ndmfv 5756 . . . . . . . . . . . . . . . 16  |-  ( -. 
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X )  ->  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )
3836, 37nsyl2 122 . . . . . . . . . . . . . . 15  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o  ->  |^|
{ a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X ) )
3938adantl 454 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( B  |`  X ) )
4039olcd 384 . . . . . . . . . . . . 13  |-  ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  ( A  |`  X )  \/  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( B  |`  X ) ) )
4124, 26, 403jaoi 1248 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )  \/  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  \/  ( ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o ) )  ->  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X )  \/  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( B  |`  X ) ) )
4215, 41sylbi 189 . . . . . . . . . . 11  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } ) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  ( A  |`  X )  \/  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( B  |`  X ) ) )
4312, 42syl6bi 221 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( A  |`  X ) < s ( B  |`  X )  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X )  \/  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( B  |`  X ) ) ) )
4443imp 420 . . . . . . . . 9  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  -> 
( |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( A  |`  X )  \/  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X ) ) )
45 dmres 5168 . . . . . . . . . . . 12  |-  dom  ( A  |`  X )  =  ( X  i^i  dom  A )
4645elin2 3532 . . . . . . . . . . 11  |-  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X )  <-> 
( |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  X  /\  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  A ) )
4746simplbi 448 . . . . . . . . . 10  |-  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  X )
48 dmres 5168 . . . . . . . . . . . 12  |-  dom  ( B  |`  X )  =  ( X  i^i  dom  B )
4948elin2 3532 . . . . . . . . . . 11  |-  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X )  <-> 
( |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  X  /\  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  B ) )
5049simplbi 448 . . . . . . . . . 10  |-  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  X )
5147, 50jaoi 370 . . . . . . . . 9  |-  ( (
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X )  \/  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( B  |`  X ) )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  X )
5244, 51syl 16 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  X )
53 onelss 4624 . . . . . . . 8  |-  ( X  e.  On  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  X  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } 
C_  X ) )
5410, 52, 53sylc 59 . . . . . . 7  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  C_  X )
5554sselda 3349 . . . . . 6  |-  ( ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) <
s ( B  |`  X ) )  /\  y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  -> 
y  e.  X )
56 onelon 4607 . . . . . . . . 9  |-  ( (
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  On  /\  y  e.  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  ->  y  e.  On )
579, 56sylan 459 . . . . . . . 8  |-  ( ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) <
s ( B  |`  X ) )  /\  y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  -> 
y  e.  On )
58 intss1 4066 . . . . . . . . . . . . 13  |-  ( y  e.  { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } 
C_  y )
59 ontri1 4616 . . . . . . . . . . . . 13  |-  ( (
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  On  /\  y  e.  On )  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  C_  y  <->  -.  y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } ) )
6058, 59syl5ib 212 . . . . . . . . . . . 12  |-  ( (
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  On  /\  y  e.  On )  ->  ( y  e. 
{ a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  ->  -.  y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } ) )
6160con2d 110 . . . . . . . . . . 11  |-  ( (
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  On  /\  y  e.  On )  ->  ( y  e. 
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  ->  -.  y  e.  { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } ) )
629, 61sylan 459 . . . . . . . . . 10  |-  ( ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) <
s ( B  |`  X ) )  /\  y  e.  On )  ->  ( y  e.  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  ->  -.  y  e.  { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } ) )
6362impancom 429 . . . . . . . . 9  |-  ( ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) <
s ( B  |`  X ) )  /\  y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  -> 
( y  e.  On  ->  -.  y  e.  {
a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) )
6457, 63mpd 15 . . . . . . . 8  |-  ( ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) <
s ( B  |`  X ) )  /\  y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  ->  -.  y  e.  { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )
65 fveq2 5729 . . . . . . . . . . . 12  |-  ( a  =  y  ->  (
( A  |`  X ) `
 a )  =  ( ( A  |`  X ) `  y
) )
66 fveq2 5729 . . . . . . . . . . . 12  |-  ( a  =  y  ->  (
( B  |`  X ) `
 a )  =  ( ( B  |`  X ) `  y
) )
6765, 66neeq12d 2617 . . . . . . . . . . 11  |-  ( a  =  y  ->  (
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a )  <->  ( ( A  |`  X ) `  y )  =/=  (
( B  |`  X ) `
 y ) ) )
6867elrab 3093 . . . . . . . . . 10  |-  ( y  e.  { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  <-> 
( y  e.  On  /\  ( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y ) ) )
6968simplbi2 610 . . . . . . . . 9  |-  ( y  e.  On  ->  (
( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y )  ->  y  e.  { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) )
7069con3d 128 . . . . . . . 8  |-  ( y  e.  On  ->  ( -.  y  e.  { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ->  -.  ( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y ) ) )
7157, 64, 70sylc 59 . . . . . . 7  |-  ( ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) <
s ( B  |`  X ) )  /\  y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  ->  -.  ( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y ) )
72 df-ne 2602 . . . . . . . 8  |-  ( ( ( A  |`  X ) `
 y )  =/=  ( ( B  |`  X ) `  y
)  <->  -.  ( ( A  |`  X ) `  y )  =  ( ( B  |`  X ) `
 y ) )
7372con2bii 324 . . . . . . 7  |-  ( ( ( A  |`  X ) `
 y )  =  ( ( B  |`  X ) `  y
)  <->  -.  ( ( A  |`  X ) `  y )  =/=  (
( B  |`  X ) `
 y ) )
7471, 73sylibr 205 . . . . . 6  |-  ( ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) <
s ( B  |`  X ) )  /\  y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  -> 
( ( A  |`  X ) `  y
)  =  ( ( B  |`  X ) `  y ) )
75 fvres 5746 . . . . . . . 8  |-  ( y  e.  X  ->  (
( A  |`  X ) `
 y )  =  ( A `  y
) )
76 fvres 5746 . . . . . . . 8  |-  ( y  e.  X  ->  (
( B  |`  X ) `
 y )  =  ( B `  y
) )
7775, 76eqeq12d 2451 . . . . . . 7  |-  ( y  e.  X  ->  (
( ( A  |`  X ) `  y
)  =  ( ( B  |`  X ) `  y )  <->  ( A `  y )  =  ( B `  y ) ) )
7877biimpd 200 . . . . . 6  |-  ( y  e.  X  ->  (
( ( A  |`  X ) `  y
)  =  ( ( B  |`  X ) `  y )  ->  ( A `  y )  =  ( B `  y ) ) )
7955, 74, 78sylc 59 . . . . 5  |-  ( ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) <
s ( B  |`  X ) )  /\  y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  -> 
( A `  y
)  =  ( B `
 y ) )
8079ralrimiva 2790 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  A. y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ( A `
 y )  =  ( B `  y
) )
81 fvresval 25392 . . . . . . . . . . . . . . 15  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  ( A `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  \/  ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )
8281ori 366 . . . . . . . . . . . . . 14  |-  ( -.  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  ->  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )
8320, 82nsyl2 122 . . . . . . . . . . . . 13  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  ->  ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  ( A `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } ) )
8483eqcomd 2442 . . . . . . . . . . . 12  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  ->  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) )
85 eqeq2 2446 . . . . . . . . . . . 12  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  ->  ( ( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  <-> 
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o ) )
8684, 85mpbid 203 . . . . . . . . . . 11  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  ->  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  1o )
8786adantr 453 . . . . . . . . . 10  |-  ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )  -> 
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o )
8887a1i 11 . . . . . . . . 9  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )  -> 
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o ) )
8922ad2antrl 710 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) ) )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( A  |`  X ) )
9089, 47syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) ) )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  X )
91 nofun 25605 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  |`  X )  e.  No  ->  Fun  ( B  |`  X ) )
92 fvelrn 5867 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  ( B  |`  X )  /\  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X ) )  ->  ( ( B  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  ran  ( B  |`  X ) )
9392ex 425 . . . . . . . . . . . . . . . . . 18  |-  ( Fun  ( B  |`  X )  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  ( B  |`  X )  ->  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  ran  ( B  |`  X ) ) )
9491, 93syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( B  |`  X )  e.  No  ->  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X )  ->  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  ran  ( B  |`  X ) ) )
95 norn 25607 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  |`  X )  e.  No  ->  ran  ( B  |`  X )  C_  { 1o ,  2o } )
9695sseld 3348 . . . . . . . . . . . . . . . . 17  |-  ( ( B  |`  X )  e.  No  ->  ( (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  ran  ( B  |`  X )  -> 
( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  { 1o ,  2o } ) )
9794, 96syld 43 . . . . . . . . . . . . . . . 16  |-  ( ( B  |`  X )  e.  No  ->  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X )  ->  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  { 1o ,  2o } ) )
98 nosgnn0 25614 . . . . . . . . . . . . . . . . 17  |-  -.  (/)  e.  { 1o ,  2o }
99 eleq1 2497 . . . . . . . . . . . . . . . . 17  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  ->  (
( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  { 1o ,  2o }  <->  (/)  e.  { 1o ,  2o } ) )
10098, 99mtbiri 296 . . . . . . . . . . . . . . . 16  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  ->  -.  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  { 1o ,  2o } )
10197, 100nsyli 136 . . . . . . . . . . . . . . 15  |-  ( ( B  |`  X )  e.  No  ->  ( (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  ->  -.  |^|
{ a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X ) ) )
1024, 101syl 16 . . . . . . . . . . . . . 14  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  ->  -.  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X ) ) )
103102imp 420 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )  ->  -.  |^|
{ a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X ) )
104103adantrl 698 . . . . . . . . . . . 12  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) ) )  ->  -.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  ( B  |`  X ) )
10549simplbi2 610 . . . . . . . . . . . . 13  |-  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  X  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  B  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( B  |`  X ) ) )
106105con3d 128 . . . . . . . . . . . 12  |-  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  X  ->  ( -.  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( B  |`  X )  ->  -.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  B ) )
10790, 104, 106sylc 59 . . . . . . . . . . 11  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) ) )  ->  -.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  B )
108 ndmfv 5756 . . . . . . . . . . 11  |-  ( -. 
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  B  ->  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )
109107, 108syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) ) )  ->  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )
110109ex 425 . . . . . . . . 9  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )  -> 
( B `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) ) )
11188, 110jcad 521 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )  -> 
( ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  (/) ) ) )
112 fvresval 25392 . . . . . . . . . . . . . 14  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  ( B `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  \/  ( ( B  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )
113112ori 366 . . . . . . . . . . . . 13  |-  ( -.  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  ->  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )
11436, 113nsyl2 122 . . . . . . . . . . . 12  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o  ->  ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  ( B `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } ) )
115114eqcomd 2442 . . . . . . . . . . 11  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o  ->  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) )
116 eqeq2 2446 . . . . . . . . . . 11  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o  ->  ( ( B `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  <-> 
( B `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  2o ) )
117115, 116mpbid 203 . . . . . . . . . 10  |-  ( ( ( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o  ->  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  2o )
11886, 117anim12i 551 . . . . . . . . 9  |-  ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  ( ( A `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  2o ) )
119118a1i 11 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  ( ( A `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  2o ) ) )
12038ad2antll 711 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o ) )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  ( B  |`  X ) )
121120, 50syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o ) )  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  X
)
122 nofun 25605 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  |`  X )  e.  No  ->  Fun  ( A  |`  X ) )
123 fvelrn 5867 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  ( A  |`  X )  /\  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X ) )  ->  ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  ran  ( A  |`  X ) )
124123ex 425 . . . . . . . . . . . . . . . . . 18  |-  ( Fun  ( A  |`  X )  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  ( A  |`  X )  ->  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  ran  ( A  |`  X ) ) )
125122, 124syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( A  |`  X )  e.  No  ->  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X )  ->  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  ran  ( A  |`  X ) ) )
126 norn 25607 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  |`  X )  e.  No  ->  ran  ( A  |`  X )  C_  { 1o ,  2o } )
127126sseld 3348 . . . . . . . . . . . . . . . . 17  |-  ( ( A  |`  X )  e.  No  ->  ( (
( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  ran  ( A  |`  X )  -> 
( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  { 1o ,  2o } ) )
128125, 127syld 43 . . . . . . . . . . . . . . . 16  |-  ( ( A  |`  X )  e.  No  ->  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X )  ->  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  { 1o ,  2o } ) )
129 eleq1 2497 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  ->  (
( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  { 1o ,  2o }  <->  (/)  e.  { 1o ,  2o } ) )
13098, 129mtbiri 296 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  ->  -.  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  e.  { 1o ,  2o } )
131128, 130nsyli 136 . . . . . . . . . . . . . . 15  |-  ( ( A  |`  X )  e.  No  ->  ( (
( A  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  ->  -.  |^|
{ a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X ) ) )
1322, 131syl 16 . . . . . . . . . . . . . 14  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  ->  -.  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X ) ) )
133132imp 420 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )  ->  -.  |^|
{ a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X ) )
134133adantrr 699 . . . . . . . . . . . 12  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o ) )  ->  -.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  ( A  |`  X ) )
13546simplbi2 610 . . . . . . . . . . . . 13  |-  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  X  ->  ( |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  A  ->  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  e.  dom  ( A  |`  X ) ) )
136135con3d 128 . . . . . . . . . . . 12  |-  ( |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  X  ->  ( -.  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  ( A  |`  X )  ->  -.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  A ) )
137121, 134, 136sylc 59 . . . . . . . . . . 11  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o ) )  ->  -.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  A )
138137ex 425 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  -.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  e.  dom  A ) )
139 ndmfv 5756 . . . . . . . . . 10  |-  ( -. 
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  dom  A  ->  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/) )
140138, 139syl6 32 . . . . . . . . 9  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) ) )
141117adantl 454 . . . . . . . . . 10  |-  ( ( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  2o )
142141a1i 11 . . . . . . . . 9  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  2o ) )
143140, 142jcad 521 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  ->  ( ( A `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  2o ) ) )
144111, 119, 1433orim123d 1263 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  1o  /\  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/) )  \/  (
( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o )  \/  ( ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  (/)  /\  (
( B  |`  X ) `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  =  2o ) )  ->  ( (
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  (/) )  \/  (
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  2o )  \/  (
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  2o ) ) ) )
145 fvex 5743 . . . . . . . 8  |-  ( A `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
146 fvex 5743 . . . . . . . 8  |-  ( B `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
147145, 146brtp 25373 . . . . . . 7  |-  ( ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  <->  ( (
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  (/) )  \/  (
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  2o )  \/  (
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  =  (/)  /\  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } )  =  2o ) ) )
148144, 15, 1473imtr4g 263 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( ( A  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( B  |`  X ) `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } )  ->  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) ) )
14912, 148sylbid 208 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( A  |`  X ) < s ( B  |`  X )  ->  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } ) ) )
150149imp 420 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  -> 
( A `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) )
151 raleq 2905 . . . . . 6  |-  ( x  =  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ->  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  <->  A. y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  ( A `  y
)  =  ( B `
 y ) ) )
152 fveq2 5729 . . . . . . 7  |-  ( x  =  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ->  ( A `  x )  =  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) )
153 fveq2 5729 . . . . . . 7  |-  ( x  =  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ->  ( B `  x )  =  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) )
154152, 153breq12d 4226 . . . . . 6  |-  ( x  =  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ->  (
( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x )  <->  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } ) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) ) )
155151, 154anbi12d 693 . . . . 5  |-  ( x  =  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ->  (
( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) )  <->  ( A. y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ( A `
 y )  =  ( B `  y
)  /\  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } ) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) ) ) )
156155rspcev 3053 . . . 4  |-  ( (
|^| { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) }  e.  On  /\  ( A. y  e.  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  ( A `  y
)  =  ( B `
 y )  /\  ( A `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) } ) ) )  ->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) )
1579, 80, 150, 156syl12anc 1183 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) )
158 sltval 25603 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
<->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) )
1591583adant3 978 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  ( A < s B  <->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) ) )
160159adantr 453 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  -> 
( A < s B 
<->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) )
161157, 160mpbird 225 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  A < s B )
162161ex 425 1  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  (
( A  |`  X ) < s ( B  |`  X )  ->  A < s B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 936    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707   {crab 2710   _Vcvv 2957    C_ wss 3321   (/)c0 3629   {cpr 3816   {ctp 3817   <.cop 3818   |^|cint 4051   class class class wbr 4213   Oncon0 4582   dom cdm 4879   ran crn 4880    |` cres 4881   Fun wfun 5449   ` cfv 5455   1oc1o 6718   2oc2o 6719   Nocsur 25596   < scslt 25597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-1o 6725  df-2o 6726  df-no 25599  df-slt 25600
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