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Theorem sltres 24318
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 24314 . . . . . . 7  |-  ( ( A  e.  No  /\  X  e.  On )  ->  ( A  |`  X )  e.  No )
213adant2 974 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  ( A  |`  X )  e.  No )
3 noreson 24314 . . . . . . 7  |-  ( ( B  e.  No  /\  X  e.  On )  ->  ( B  |`  X )  e.  No )
433adant1 973 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  ( B  |`  X )  e.  No )
5 sltintdifex 24317 . . . . . . 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 4592 . . . . . . 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 217 . . . . . 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 642 . . . . 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 418 . . . 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 960 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  X  e.  On )
11 sltval2 24310 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 5539 . . . . . . . . . . . . 13  |-  ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
14 fvex 5539 . . . . . . . . . . . . 13  |-  ( ( B  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
1513, 14brtp 24106 . . . . . . . . . . . 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 6494 . . . . . . . . . . . . . . . . . 18  |-  1o  =/=  (/)
17 df-ne 2448 . . . . . . . . . . . . . . . . . 18  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
1816, 17mpbi 199 . . . . . . . . . . . . . . . . 17  |-  -.  1o  =  (/)
19 eqeq1 2289 . . . . . . . . . . . . . . . . 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 294 . . . . . . . . . . . . . . . 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 5552 . . . . . . . . . . . . . . . 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 119 . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 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 381 . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 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 381 . . . . . . . . . . . . 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 6487 . . . . . . . . . . . . . . . . . . . . 21  |-  2o  e.  On
2827elexi 2797 . . . . . . . . . . . . . . . . . . . 20  |-  2o  e.  _V
2928prid2 3735 . . . . . . . . . . . . . . . . . . 19  |-  2o  e.  { 1o ,  2o }
3029nosgnn0i 24313 . . . . . . . . . . . . . . . . . 18  |-  (/)  =/=  2o
31 df-ne 2448 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  =/=  2o  <->  -.  (/)  =  2o )
3230, 31mpbi 199 . . . . . . . . . . . . . . . . 17  |-  -.  (/)  =  2o
33 eqeq1 2289 . . . . . . . . . . . . . . . . . 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 2285 . . . . . . . . . . . . . . . . . 18  |-  ( 2o  =  (/)  <->  (/)  =  2o )
3533, 34syl6bb 252 . . . . . . . . . . . . . . . . 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 294 . . . . . . . . . . . . . . . 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 5552 . . . . . . . . . . . . . . . 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 119 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 382 . . . . . . . . . . . . 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 1245 . . . . . . . . . . . 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 187 . . . . . . . . . . 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 219 . . . . . . . . . 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 418 . . . . . . . . 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 4976 . . . . . . . . . . . 12  |-  dom  ( A  |`  X )  =  ( X  i^i  dom  A )
4645elin2 3359 . . . . . . . . . . 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 446 . . . . . . . . . 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 4976 . . . . . . . . . . . 12  |-  dom  ( B  |`  X )  =  ( X  i^i  dom  B )
4948elin2 3359 . . . . . . . . . . 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 446 . . . . . . . . . 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 368 . . . . . . . . 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 15 . . . . . . . 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 4434 . . . . . . . 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 56 . . . . . . 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 3180 . . . . . 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 4417 . . . . . . . . 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 457 . . . . . . . 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 3877 . . . . . . . . . . . . 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 4426 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 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 107 . . . . . . . . . . 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 457 . . . . . . . . . 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 427 . . . . . . . . 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 14 . . . . . . . 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 5525 . . . . . . . . . . . 12  |-  ( a  =  y  ->  (
( A  |`  X ) `
 a )  =  ( ( A  |`  X ) `  y
) )
66 fveq2 5525 . . . . . . . . . . . 12  |-  ( a  =  y  ->  (
( B  |`  X ) `
 a )  =  ( ( B  |`  X ) `  y
) )
6765, 66neeq12d 2461 . . . . . . . . . . 11  |-  ( a  =  y  ->  (
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a )  <->  ( ( A  |`  X ) `  y )  =/=  (
( B  |`  X ) `
 y ) ) )
6867elrab 2923 . . . . . . . . . 10  |-  ( y  e.  { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  <-> 
( y  e.  On  /\  ( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y ) ) )
6968simplbi2 608 . . . . . . . . 9  |-  ( y  e.  On  ->  (
( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y )  ->  y  e.  { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) )
7069con3d 125 . . . . . . . 8  |-  ( y  e.  On  ->  ( -.  y  e.  { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ->  -.  ( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y ) ) )
7157, 64, 70sylc 56 . . . . . . 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 2448 . . . . . . . 8  |-  ( ( ( A  |`  X ) `
 y )  =/=  ( ( B  |`  X ) `  y
)  <->  -.  ( ( A  |`  X ) `  y )  =  ( ( B  |`  X ) `
 y ) )
7372con2bii 322 . . . . . . 7  |-  ( ( ( A  |`  X ) `
 y )  =  ( ( B  |`  X ) `  y
)  <->  -.  ( ( A  |`  X ) `  y )  =/=  (
( B  |`  X ) `
 y ) )
7471, 73sylibr 203 . . . . . 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 5542 . . . . . . . 8  |-  ( y  e.  X  ->  (
( A  |`  X ) `
 y )  =  ( A `  y
) )
76 fvres 5542 . . . . . . . 8  |-  ( y  e.  X  ->  (
( B  |`  X ) `
 y )  =  ( B `  y
) )
7775, 76eqeq12d 2297 . . . . . . 7  |-  ( y  e.  X  ->  (
( ( A  |`  X ) `  y
)  =  ( ( B  |`  X ) `  y )  <->  ( A `  y )  =  ( B `  y ) ) )
7877biimpd 198 . . . . . 6  |-  ( y  e.  X  ->  (
( ( A  |`  X ) `  y
)  =  ( ( B  |`  X ) `  y )  ->  ( A `  y )  =  ( B `  y ) ) )
7955, 74, 78sylc 56 . . . . 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 2626 . . . 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 24123 . . . . . . . . . . . . . . 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 364 . . . . . . . . . . . . . 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 119 . . . . . . . . . . . . 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 2288 . . . . . . . . . . . 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 2292 . . . . . . . . . . . 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 201 . . . . . . . . . . 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 451 . . . . . . . . . 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 10 . . . . . . . . 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 708 . . . . . . . . . . . . 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 15 . . . . . . . . . . . 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 24303 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  |`  X )  e.  No  ->  Fun  ( B  |`  X ) )
92 fvelrn 5661 . . . . . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . . 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 24305 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  |`  X )  e.  No  ->  ran  ( B  |`  X )  C_  { 1o ,  2o } )
9695sseld 3179 . . . . . . . . . . . . . . . . 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 40 . . . . . . . . . . . . . . . 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 24312 . . . . . . . . . . . . . . . . 17  |-  -.  (/)  e.  { 1o ,  2o }
99 eleq1 2343 . . . . . . . . . . . . . . . . 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 294 . . . . . . . . . . . . . . . 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 133 . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . 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 418 . . . . . . . . . . . . 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 696 . . . . . . . . . . . 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 608 . . . . . . . . . . . . 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 125 . . . . . . . . . . . 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 56 . . . . . . . . . . 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 5552 . . . . . . . . . . 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 15 . . . . . . . . . 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 423 . . . . . . . . 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 519 . . . . . . . 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 24123 . . . . . . . . . . . . . 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 364 . . . . . . . . . . . . 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 119 . . . . . . . . . . . 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 2288 . . . . . . . . . . 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 2292 . . . . . . . . . . 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 201 . . . . . . . . . 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 549 . . . . . . . . 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 10 . . . . . . . 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 709 . . . . . . . . . . . . 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 15 . . . . . . . . . . . 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 24303 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  |`  X )  e.  No  ->  Fun  ( A  |`  X ) )
123 fvelrn 5661 . . . . . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . . 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 24305 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  |`  X )  e.  No  ->  ran  ( A  |`  X )  C_  { 1o ,  2o } )
127126sseld 3179 . . . . . . . . . . . . . . . . 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 40 . . . . . . . . . . . . . . . 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 2343 . . . . . . . . . . . . . . . . 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 294 . . . . . . . . . . . . . . . 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 133 . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . 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 418 . . . . . . . . . . . . 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 697 . . . . . . . . . . . 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 608 . . . . . . . . . . . . 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 125 . . . . . . . . . . . 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 56 . . . . . . . . . . 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 423 . . . . . . . . . 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 5552 . . . . . . . . . 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 29 . . . . . . . . 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 452 . . . . . . . . . 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 10 . . . . . . . . 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 519 . . . . . . . 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 1260 . . . . . . 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 5539 . . . . . . . 8  |-  ( A `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
146 fvex 5539 . . . . . . . 8  |-  ( B `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
147145, 146brtp 24106 . . . . . . 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 261 . . . . . 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 206 . . . . 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 418 . . . 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 2736 . . . . . 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 5525 . . . . . . 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 5525 . . . . . . 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 4036 . . . . . 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 691 . . . . 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 2884 . . . 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 1180 . . 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 24301 . . . . 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 975 . . . 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 451 . . 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 223 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  A < s B )
162161ex 423 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 176    \/ wo 357    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {cpr 3641   {ctp 3642   <.cop 3643   |^|cint 3862   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ran crn 4690    |` cres 4691   Fun wfun 5249   ` cfv 5255   1oc1o 6472   2oc2o 6473   Nocsur 24294   < scslt 24295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-2o 6480  df-no 24297  df-slt 24298
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