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Theorem sltres 25532
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 25528 . . . . . . 7  |-  ( ( A  e.  No  /\  X  e.  On )  ->  ( A  |`  X )  e.  No )
213adant2 976 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  ( A  |`  X )  e.  No )
3 noreson 25528 . . . . . . 7  |-  ( ( B  e.  No  /\  X  e.  On )  ->  ( B  |`  X )  e.  No )
433adant1 975 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  ( B  |`  X )  e.  No )
5 sltintdifex 25531 . . . . . . 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 4740 . . . . . . 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 218 . . . . . 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 643 . . . . 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 419 . . . 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 962 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  X  e.  On )
11 sltval2 25524 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 5701 . . . . . . . . . . . . 13  |-  ( ( A  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
14 fvex 5701 . . . . . . . . . . . . 13  |-  ( ( B  |`  X ) `  |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
1513, 14brtp 25320 . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . . 18  |-  1o  =/=  (/)
17 df-ne 2569 . . . . . . . . . . . . . . . . . 18  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
1816, 17mpbi 200 . . . . . . . . . . . . . . . . 17  |-  -.  1o  =  (/)
19 eqeq1 2410 . . . . . . . . . . . . . . . . 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 295 . . . . . . . . . . . . . . . 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 5714 . . . . . . . . . . . . . . . 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 121 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 382 . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 382 . . . . . . . . . . . . 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 6691 . . . . . . . . . . . . . . . . . . . . 21  |-  2o  e.  On
2827elexi 2925 . . . . . . . . . . . . . . . . . . . 20  |-  2o  e.  _V
2928prid2 3873 . . . . . . . . . . . . . . . . . . 19  |-  2o  e.  { 1o ,  2o }
3029nosgnn0i 25527 . . . . . . . . . . . . . . . . . 18  |-  (/)  =/=  2o
31 df-ne 2569 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  =/=  2o  <->  -.  (/)  =  2o )
3230, 31mpbi 200 . . . . . . . . . . . . . . . . 17  |-  -.  (/)  =  2o
33 eqeq1 2410 . . . . . . . . . . . . . . . . . 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 2406 . . . . . . . . . . . . . . . . . 18  |-  ( 2o  =  (/)  <->  (/)  =  2o )
3533, 34syl6bb 253 . . . . . . . . . . . . . . . . 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 295 . . . . . . . . . . . . . . . 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 5714 . . . . . . . . . . . . . . . 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 121 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 383 . . . . . . . . . . . . 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 1247 . . . . . . . . . . . 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 188 . . . . . . . . . . 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 220 . . . . . . . . . 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 419 . . . . . . . . 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 5126 . . . . . . . . . . . 12  |-  dom  ( A  |`  X )  =  ( X  i^i  dom  A )
4645elin2 3491 . . . . . . . . . . 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 447 . . . . . . . . . 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 5126 . . . . . . . . . . . 12  |-  dom  ( B  |`  X )  =  ( X  i^i  dom  B )
4948elin2 3491 . . . . . . . . . . 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 447 . . . . . . . . . 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 369 . . . . . . . . 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 4583 . . . . . . . 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 58 . . . . . . 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 3308 . . . . . 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 4566 . . . . . . . . 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 458 . . . . . . . 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 4025 . . . . . . . . . . . . 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 4575 . . . . . . . . . . . . 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 211 . . . . . . . . . . . 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 109 . . . . . . . . . . 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 458 . . . . . . . . . 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 428 . . . . . . . . 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 5687 . . . . . . . . . . . 12  |-  ( a  =  y  ->  (
( A  |`  X ) `
 a )  =  ( ( A  |`  X ) `  y
) )
66 fveq2 5687 . . . . . . . . . . . 12  |-  ( a  =  y  ->  (
( B  |`  X ) `
 a )  =  ( ( B  |`  X ) `  y
) )
6765, 66neeq12d 2582 . . . . . . . . . . 11  |-  ( a  =  y  ->  (
( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a )  <->  ( ( A  |`  X ) `  y )  =/=  (
( B  |`  X ) `
 y ) ) )
6867elrab 3052 . . . . . . . . . 10  |-  ( y  e.  { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) }  <-> 
( y  e.  On  /\  ( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y ) ) )
6968simplbi2 609 . . . . . . . . 9  |-  ( y  e.  On  ->  (
( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y )  ->  y  e.  { a  e.  On  |  ( ( A  |`  X ) `  a
)  =/=  ( ( B  |`  X ) `  a ) } ) )
7069con3d 127 . . . . . . . 8  |-  ( y  e.  On  ->  ( -.  y  e.  { a  e.  On  |  ( ( A  |`  X ) `
 a )  =/=  ( ( B  |`  X ) `  a
) }  ->  -.  ( ( A  |`  X ) `  y
)  =/=  ( ( B  |`  X ) `  y ) ) )
7157, 64, 70sylc 58 . . . . . . 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 2569 . . . . . . . 8  |-  ( ( ( A  |`  X ) `
 y )  =/=  ( ( B  |`  X ) `  y
)  <->  -.  ( ( A  |`  X ) `  y )  =  ( ( B  |`  X ) `
 y ) )
7372con2bii 323 . . . . . . 7  |-  ( ( ( A  |`  X ) `
 y )  =  ( ( B  |`  X ) `  y
)  <->  -.  ( ( A  |`  X ) `  y )  =/=  (
( B  |`  X ) `
 y ) )
7471, 73sylibr 204 . . . . . 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 5704 . . . . . . . 8  |-  ( y  e.  X  ->  (
( A  |`  X ) `
 y )  =  ( A `  y
) )
76 fvres 5704 . . . . . . . 8  |-  ( y  e.  X  ->  (
( B  |`  X ) `
 y )  =  ( B `  y
) )
7775, 76eqeq12d 2418 . . . . . . 7  |-  ( y  e.  X  ->  (
( ( A  |`  X ) `  y
)  =  ( ( B  |`  X ) `  y )  <->  ( A `  y )  =  ( B `  y ) ) )
7877biimpd 199 . . . . . 6  |-  ( y  e.  X  ->  (
( ( A  |`  X ) `  y
)  =  ( ( B  |`  X ) `  y )  ->  ( A `  y )  =  ( B `  y ) ) )
7955, 74, 78sylc 58 . . . . 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 2749 . . . 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 25337 . . . . . . . . . . . . . . 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 365 . . . . . . . . . . . . . 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 121 . . . . . . . . . . . . 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 2409 . . . . . . . . . . . 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 2413 . . . . . . . . . . . 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 202 . . . . . . . . . . 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 452 . . . . . . . . . 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 709 . . . . . . . . . . . . 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 25517 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  |`  X )  e.  No  ->  Fun  ( B  |`  X ) )
92 fvelrn 5825 . . . . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . . . . 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 25519 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  |`  X )  e.  No  ->  ran  ( B  |`  X )  C_  { 1o ,  2o } )
9695sseld 3307 . . . . . . . . . . . . . . . . 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 42 . . . . . . . . . . . . . . . 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 25526 . . . . . . . . . . . . . . . . 17  |-  -.  (/)  e.  { 1o ,  2o }
99 eleq1 2464 . . . . . . . . . . . . . . . . 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 295 . . . . . . . . . . . . . . . 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 135 . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . 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 697 . . . . . . . . . . . 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 609 . . . . . . . . . . . . 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 127 . . . . . . . . . . . 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 58 . . . . . . . . . . 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 5714 . . . . . . . . . . 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 424 . . . . . . . . 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 520 . . . . . . . 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 25337 . . . . . . . . . . . . . 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 365 . . . . . . . . . . . . 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 121 . . . . . . . . . . . 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 2409 . . . . . . . . . . 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 2413 . . . . . . . . . . 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 202 . . . . . . . . . 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 550 . . . . . . . . 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 710 . . . . . . . . . . . . 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 25517 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  |`  X )  e.  No  ->  Fun  ( A  |`  X ) )
123 fvelrn 5825 . . . . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . . . . 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 25519 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  |`  X )  e.  No  ->  ran  ( A  |`  X )  C_  { 1o ,  2o } )
127126sseld 3307 . . . . . . . . . . . . . . . . 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 42 . . . . . . . . . . . . . . . 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 2464 . . . . . . . . . . . . . . . . 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 295 . . . . . . . . . . . . . . . 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 135 . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . 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 698 . . . . . . . . . . . 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 609 . . . . . . . . . . . . 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 127 . . . . . . . . . . . 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 58 . . . . . . . . . . 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 424 . . . . . . . . . 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 5714 . . . . . . . . . 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 31 . . . . . . . . 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 453 . . . . . . . . . 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 520 . . . . . . . 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 1262 . . . . . . 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 5701 . . . . . . . 8  |-  ( A `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
146 fvex 5701 . . . . . . . 8  |-  ( B `
 |^| { a  e.  On  |  ( ( A  |`  X ) `  a )  =/=  (
( B  |`  X ) `
 a ) } )  e.  _V
147145, 146brtp 25320 . . . . . . 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 262 . . . . . 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 207 . . . . 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 419 . . . 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 2864 . . . . . 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 5687 . . . . . . 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 5687 . . . . . . 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 4185 . . . . . 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 692 . . . . 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 3012 . . . 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 1182 . . 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 25515 . . . . 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 977 . . . 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 452 . . 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 224 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  /\  ( A  |`  X ) < s ( B  |`  X ) )  ->  A < s B )
162161ex 424 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 177    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916    C_ wss 3280   (/)c0 3588   {cpr 3775   {ctp 3776   <.cop 3777   |^|cint 4010   class class class wbr 4172   Oncon0 4541   dom cdm 4837   ran crn 4838    |` cres 4839   Fun wfun 5407   ` cfv 5413   1oc1o 6676   2oc2o 6677   Nocsur 25508   < scslt 25509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-2o 6684  df-no 25511  df-slt 25512
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