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Theorem sltintdifex 24317
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 24310 . 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 5539 . . . 4  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
3 fvex 5539 . . . 4  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
42, 3brtp 24106 . . 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 5519 . . . . . . 7  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
6 1n0 6494 . . . . . . . . 9  |-  1o  =/=  (/)
7 df-ne 2448 . . . . . . . . 9  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
86, 7mpbi 199 . . . . . . . 8  |-  -.  1o  =  (/)
9 eqeq1 2289 . . . . . . . . 9  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  <->  (/)  =  1o ) )
10 eqcom 2285 . . . . . . . . 9  |-  ( (/)  =  1o  <->  1o  =  (/) )
119, 10syl6bb 252 . . . . . . . 8  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  <->  1o  =  (/) ) )
128, 11mtbiri 294 . . . . . . 7  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  -.  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o )
135, 12syl 15 . . . . . 6  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  -.  ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  1o )
1413con4i 122 . . . . 5  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
1514adantr 451 . . . 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 451 . . . 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 5519 . . . . . . 7  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
18 2on0 6488 . . . . . . . . 9  |-  2o  =/=  (/)
19 df-ne 2448 . . . . . . . . 9  |-  ( 2o  =/=  (/)  <->  -.  2o  =  (/) )
2018, 19mpbi 199 . . . . . . . 8  |-  -.  2o  =  (/)
21 eqeq1 2289 . . . . . . . . 9  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  <->  (/)  =  2o ) )
22 eqcom 2285 . . . . . . . . 9  |-  ( (/)  =  2o  <->  2o  =  (/) )
2321, 22syl6bb 252 . . . . . . . 8  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  <->  2o  =  (/) ) )
2420, 23mtbiri 294 . . . . . . 7  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )
2517, 24syl 15 . . . . . 6  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  -.  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  2o )
2625con4i 122 . . . . 5  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
2726adantl 452 . . . 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 1245 . . 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 187 . 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 219 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 358    \/ w3o 933    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788   (/)c0 3455   {ctp 3642   <.cop 3643   |^|cint 3862   class class class wbr 4023   Oncon0 4392   ` cfv 5255   1oc1o 6472   2oc2o 6473   Nocsur 24294   < scslt 24295
This theorem is referenced by:  sltres  24318
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-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-rab 2552  df-v 2790  df-sbc 2992  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-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-iota 5219  df-fv 5263  df-1o 6479  df-2o 6480  df-slt 24298
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