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Theorem sltintdifex 24875
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 24868 . 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 5619 . . . 4  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
3 fvex 5619 . . . 4  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
42, 3brtp 24664 . . 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 5599 . . . . . . 7  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
6 1n0 6578 . . . . . . . . 9  |-  1o  =/=  (/)
7 df-ne 2523 . . . . . . . . 9  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
86, 7mpbi 199 . . . . . . . 8  |-  -.  1o  =  (/)
9 eqeq1 2364 . . . . . . . . 9  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  <->  (/)  =  1o ) )
10 eqcom 2360 . . . . . . . . 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 5599 . . . . . . 7  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
18 2on0 6572 . . . . . . . . 9  |-  2o  =/=  (/)
19 df-ne 2523 . . . . . . . . 9  |-  ( 2o  =/=  (/)  <->  -.  2o  =  (/) )
2018, 19mpbi 199 . . . . . . . 8  |-  -.  2o  =  (/)
21 eqeq1 2364 . . . . . . . . 9  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  <->  (/)  =  2o ) )
22 eqcom 2360 . . . . . . . . 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 1642    e. wcel 1710    =/= wne 2521   {crab 2623   _Vcvv 2864   (/)c0 3531   {ctp 3718   <.cop 3719   |^|cint 3941   class class class wbr 4102   Oncon0 4471   ` cfv 5334   1oc1o 6556   2oc2o 6557   Nocsur 24852   < scslt 24853
This theorem is referenced by:  sltres  24876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-br 4103  df-opab 4157  df-tr 4193  df-eprel 4384  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-suc 4477  df-iota 5298  df-fv 5342  df-1o 6563  df-2o 6564  df-slt 24856
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