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Theorem sltintdifex 25623
Description: If  A < s B, then the intersection of all the ordinals that have differing signs in  A and  B exists. (Contributed by Scott Fenton, 22-Feb-2012.)
Assertion
Ref Expression
sltintdifex  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  _V ) )
Distinct variable groups:    A, a    B, a

Proof of Theorem sltintdifex
StepHypRef Expression
1 sltval2 25616 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
<->  ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
2 fvex 5745 . . . 4  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
3 fvex 5745 . . . 4  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
42, 3brtp 25377 . . 3  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  <->  ( ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )  \/  (
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )  \/  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  /\  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  2o ) ) )
5 fvprc 5725 . . . . . . 7  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
6 1n0 6742 . . . . . . . . 9  |-  1o  =/=  (/)
7 df-ne 2603 . . . . . . . . 9  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
86, 7mpbi 201 . . . . . . . 8  |-  -.  1o  =  (/)
9 eqeq1 2444 . . . . . . . . 9  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  <->  (/)  =  1o ) )
10 eqcom 2440 . . . . . . . . 9  |-  ( (/)  =  1o  <->  1o  =  (/) )
119, 10syl6bb 254 . . . . . . . 8  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  <->  1o  =  (/) ) )
128, 11mtbiri 296 . . . . . . 7  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  -.  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o )
135, 12syl 16 . . . . . 6  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  -.  ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  1o )
1413con4i 125 . . . . 5  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
1514adantr 453 . . . 4  |-  ( ( ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
1614adantr 453 . . . 4  |-  ( ( ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
17 fvprc 5725 . . . . . . 7  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
18 2on0 6736 . . . . . . . . 9  |-  2o  =/=  (/)
19 df-ne 2603 . . . . . . . . 9  |-  ( 2o  =/=  (/)  <->  -.  2o  =  (/) )
2018, 19mpbi 201 . . . . . . . 8  |-  -.  2o  =  (/)
21 eqeq1 2444 . . . . . . . . 9  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  <->  (/)  =  2o ) )
22 eqcom 2440 . . . . . . . . 9  |-  ( (/)  =  2o  <->  2o  =  (/) )
2321, 22syl6bb 254 . . . . . . . 8  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  <->  2o  =  (/) ) )
2420, 23mtbiri 296 . . . . . . 7  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )
2517, 24syl 16 . . . . . 6  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  -.  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  2o )
2625con4i 125 . . . . 5  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
2726adantl 454 . . . 4  |-  ( ( ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
2815, 16, 273jaoi 1248 . . 3  |-  ( ( ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o 
/\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )  \/  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )  \/  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  /\  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  2o ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
294, 28sylbi 189 . 2  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
301, 29syl6bi 221 1  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   _Vcvv 2958   (/)c0 3630   {ctp 3818   <.cop 3819   |^|cint 4052   class class class wbr 4215   Oncon0 4584   ` cfv 5457   1oc1o 6720   2oc2o 6721   Nocsur 25600   < scslt 25601
This theorem is referenced by:  sltres  25624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-suc 4590  df-iota 5421  df-fv 5465  df-1o 6727  df-2o 6728  df-slt 25604
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