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Theorem sltval2 25603
Description: Alternate expression for surreal less than. Two surreals obey surreal less than iff they obey the sign ordering at the first place they differ. (Contributed by Scott Fenton, 17-Jun-2011.)
Assertion
Ref Expression
sltval2  |-  ( ( 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 ) } ) ) )
Distinct variable groups:    A, a    B, a

Proof of Theorem sltval2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sltval 25594 . 2  |-  ( ( 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
) ) ) )
2 fvex 5734 . . . . . . . . . . . . 13  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
3 fvex 5734 . . . . . . . . . . . . 13  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
42, 3brtp 25364 . . . . . . . . . . . 12  |-  ( ( 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 1n0 6731 . . . . . . . . . . . . . . . . 17  |-  1o  =/=  (/)
6 df-ne 2600 . . . . . . . . . . . . . . . . 17  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
75, 6mpbi 200 . . . . . . . . . . . . . . . 16  |-  -.  1o  =  (/)
8 eqeq1 2441 . . . . . . . . . . . . . . . 16  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  (
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  <->  1o  =  (/) ) )
97, 8mtbiri 295 . . . . . . . . . . . . . . 15  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  -.  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
10 fvprc 5714 . . . . . . . . . . . . . . 15  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
119, 10nsyl2 121 . . . . . . . . . . . . . 14  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
1211adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( 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 )
1311adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( 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 )
14 2on0 6725 . . . . . . . . . . . . . . . . 17  |-  2o  =/=  (/)
15 df-ne 2600 . . . . . . . . . . . . . . . . 17  |-  ( 2o  =/=  (/)  <->  -.  2o  =  (/) )
1614, 15mpbi 200 . . . . . . . . . . . . . . . 16  |-  -.  2o  =  (/)
17 eqeq1 2441 . . . . . . . . . . . . . . . 16  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  (
( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  <->  2o  =  (/) ) )
1816, 17mtbiri 295 . . . . . . . . . . . . . . 15  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
19 fvprc 5714 . . . . . . . . . . . . . . 15  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
2018, 19nsyl2 121 . . . . . . . . . . . . . 14  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
2120adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( 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 )
2212, 13, 213jaoi 1247 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )
234, 22sylbi 188 . . . . . . . . . . 11  |-  ( ( 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 )
24 onintrab 4773 . . . . . . . . . . 11  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  _V  <->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
2523, 24sylib 189 . . . . . . . . . 10  |-  ( ( 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.  On )
2625adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( 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.  On )
27 onelon 4598 . . . . . . . . . . . . . . 15  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  -> 
y  e.  On )
2827expcom 425 . . . . . . . . . . . . . 14  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  ->  y  e.  On ) )
2926, 28syl5 30 . . . . . . . . . . . . 13  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  (
( ( A  e.  No  /\  B  e.  No )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) )  ->  y  e.  On ) )
30 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( a  =  y  ->  ( A `  a )  =  ( A `  y ) )
31 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( a  =  y  ->  ( B `  a )  =  ( B `  y ) )
3230, 31neeq12d 2613 . . . . . . . . . . . . . . 15  |-  ( a  =  y  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  y )  =/=  ( B `  y )
) )
3332onnminsb 4776 . . . . . . . . . . . . . 14  |-  ( y  e.  On  ->  (
y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  -.  ( A `  y
)  =/=  ( B `
 y ) ) )
3433com12 29 . . . . . . . . . . . . 13  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  (
y  e.  On  ->  -.  ( A `  y
)  =/=  ( B `
 y ) ) )
3529, 34syld 42 . . . . . . . . . . . 12  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  (
( ( A  e.  No  /\  B  e.  No )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) )  ->  -.  ( A `  y )  =/=  ( B `  y ) ) )
3635com12 29 . . . . . . . . . . 11  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  -> 
( y  e.  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  ->  -.  ( A `  y )  =/=  ( B `  y )
) )
37 df-ne 2600 . . . . . . . . . . . 12  |-  ( ( A `  y )  =/=  ( B `  y )  <->  -.  ( A `  y )  =  ( B `  y ) )
3837con2bii 323 . . . . . . . . . . 11  |-  ( ( A `  y )  =  ( B `  y )  <->  -.  ( A `  y )  =/=  ( B `  y
) )
3936, 38syl6ibr 219 . . . . . . . . . 10  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  -> 
( y  e.  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  ->  ( A `  y )  =  ( B `  y ) ) )
4039ralrimiv 2780 . . . . . . . . 9  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  ->  A. y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( B `  y ) )
4126, 40jca 519 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( 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.  On  /\ 
A. y  e.  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  ( A `  y
)  =  ( B `
 y ) ) )
4241ex 424 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( 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.  On  /\  A. y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( B `  y ) ) ) )
4342impac 605 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( 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.  On  /\  A. y  e. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( B `  y ) )  /\  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
44 anass 631 . . . . . 6  |-  ( ( ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  A. y  e. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( B `  y ) )  /\  ( 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.  On  /\  ( A. y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ( A `
 y )  =  ( B `  y
)  /\  ( A `  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) ) )
4543, 44sylib 189 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( 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.  On  /\  ( A. y  e. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( B `  y )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) ) )
46 raleq 2896 . . . . . . 7  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  <->  A. y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ( A `
 y )  =  ( B `  y
) ) )
47 fveq2 5720 . . . . . . . 8  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  x )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
48 fveq2 5720 . . . . . . . 8  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  x )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
4947, 48breq12d 4217 . . . . . . 7  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x )  <->  ( A `  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
5046, 49anbi12d 692 . . . . . 6  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) )  <->  ( A. y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( B `  y )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) ) )
5150rspcev 3044 . . . . 5  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  ( A. y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( B `  y )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) )  ->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) )
5245, 51syl 16 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  ->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) )
5352ex 424 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) ) )
54 eqeq12 2447 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  (/) )  ->  (
( A `  x
)  =  ( B `
 x )  <->  1o  =  (/) ) )
557, 54mtbiri 295 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  (/) )  ->  -.  ( A `  x )  =  ( B `  x ) )
56 1on 6723 . . . . . . . . . . . . . . . . 17  |-  1o  e.  On
57 0elon 4626 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
58 suc11 4677 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =  suc  (/)  <->  1o  =  (/) ) )
5958necon3bid 2633 . . . . . . . . . . . . . . . . 17  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =/=  suc  (/) 
<->  1o  =/=  (/) ) )
6056, 57, 59mp2an 654 . . . . . . . . . . . . . . . 16  |-  ( suc 
1o  =/=  suc  (/)  <->  1o  =/=  (/) )
615, 60mpbir 201 . . . . . . . . . . . . . . 15  |-  suc  1o  =/=  suc  (/)
62 df-2o 6717 . . . . . . . . . . . . . . . 16  |-  2o  =  suc  1o
63 df-1o 6716 . . . . . . . . . . . . . . . 16  |-  1o  =  suc  (/)
6462, 63eqeq12i 2448 . . . . . . . . . . . . . . 15  |-  ( 2o  =  1o  <->  suc  1o  =  suc  (/) )
6561, 64nemtbir 2686 . . . . . . . . . . . . . 14  |-  -.  2o  =  1o
66 eqeq12 2447 . . . . . . . . . . . . . . 15  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  -> 
( ( A `  x )  =  ( B `  x )  <-> 
1o  =  2o ) )
67 eqcom 2437 . . . . . . . . . . . . . . 15  |-  ( 1o  =  2o  <->  2o  =  1o )
6866, 67syl6bb 253 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  -> 
( ( A `  x )  =  ( B `  x )  <-> 
2o  =  1o ) )
6965, 68mtbiri 295 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  ->  -.  ( A `  x
)  =  ( B `
 x ) )
70 eqcom 2437 . . . . . . . . . . . . . . 15  |-  ( 2o  =  (/)  <->  (/)  =  2o )
7116, 70mtbi 290 . . . . . . . . . . . . . 14  |-  -.  (/)  =  2o
72 eqeq12 2447 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  (/)  /\  ( B `  x )  =  2o )  ->  (
( A `  x
)  =  ( B `
 x )  <->  (/)  =  2o ) )
7371, 72mtbiri 295 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  (/)  /\  ( B `  x )  =  2o )  ->  -.  ( A `  x )  =  ( B `  x ) )
7455, 69, 733jaoi 1247 . . . . . . . . . . . 12  |-  ( ( ( ( A `  x )  =  1o 
/\  ( B `  x )  =  (/) )  \/  ( ( A `  x )  =  1o  /\  ( B `  x )  =  2o )  \/  (
( A `  x
)  =  (/)  /\  ( B `  x )  =  2o ) )  ->  -.  ( A `  x
)  =  ( B `
 x ) )
75 fvex 5734 . . . . . . . . . . . . 13  |-  ( A `
 x )  e. 
_V
76 fvex 5734 . . . . . . . . . . . . 13  |-  ( B `
 x )  e. 
_V
7775, 76brtp 25364 . . . . . . . . . . . 12  |-  ( ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
)  <->  ( ( ( A `  x )  =  1o  /\  ( B `  x )  =  (/) )  \/  (
( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  \/  ( ( A `  x )  =  (/)  /\  ( B `  x
)  =  2o ) ) )
78 df-ne 2600 . . . . . . . . . . . 12  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
7974, 77, 783imtr4i 258 . . . . . . . . . . 11  |-  ( ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
)  ->  ( A `  x )  =/=  ( B `  x )
)
80 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( A `  a )  =  ( A `  x ) )
81 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( B `  a )  =  ( B `  x ) )
8280, 81neeq12d 2613 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  x )  =/=  ( B `  x )
) )
8382elrab 3084 . . . . . . . . . . . . . 14  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  <->  ( x  e.  On  /\  ( A `
 x )  =/=  ( B `  x
) ) )
8483biimpri 198 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  ( A `  x )  =/=  ( B `  x ) )  ->  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
8584adantlr 696 . . . . . . . . . . . 12  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  ( A `  x )  =/=  ( B `  x ) )  ->  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
86 ssrab2 3420 . . . . . . . . . . . . . . . . . 18  |-  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On
87 ne0i 3626 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  =/=  (/) )
8887adantl 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  =/=  (/) )
89 onint 4767 . . . . . . . . . . . . . . . . . 18  |-  ( ( { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On  /\  {
a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  =/=  (/) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
9086, 88, 89sylancr 645 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
91 nfrab1 2880 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
9291nfint 4052 . . . . . . . . . . . . . . . . . . 19  |-  F/_ a |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
93 nfcv 2571 . . . . . . . . . . . . . . . . . . 19  |-  F/_ a On
94 nfcv 2571 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ a A
9594, 92nffv 5727 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
96 nfcv 2571 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ a B
9796, 92nffv 5727 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a
( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
9895, 97nfne 2689 . . . . . . . . . . . . . . . . . . 19  |-  F/ a ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =/=  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
99 fveq2 5720 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  a )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
100 fveq2 5720 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  a )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
10199, 100neeq12d 2613 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( A `  a
)  =/=  ( B `
 a )  <->  ( A `  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  =/=  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) )
10292, 93, 98, 101elrabf 3083 . . . . . . . . . . . . . . . . . 18  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  <->  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  On  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =/=  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
103102simprbi 451 . . . . . . . . . . . . . . . . 17  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =/=  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
10490, 103syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  -> 
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =/=  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) )
105 df-ne 2600 . . . . . . . . . . . . . . . 16  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =/=  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  <->  -.  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
106104, 105sylib 189 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  -.  ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) )
107 fveq2 5720 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  y )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
108 fveq2 5720 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  y )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
109107, 108eqeq12d 2449 . . . . . . . . . . . . . . . . 17  |-  ( y  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( A `  y
)  =  ( B `
 y )  <->  ( A `  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  =  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) )
110109rspccv 3041 . . . . . . . . . . . . . . . 16  |-  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  x  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
111110ad2antlr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  -> 
( |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  x  ->  ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) ) )
112106, 111mtod 170 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  -.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  x
)
113 simpll 731 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  x  e.  On )
114 oninton 4772 . . . . . . . . . . . . . . . . 17  |-  ( ( { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On  /\  {
a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  =/=  (/) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
11586, 87, 114sylancr 645 . . . . . . . . . . . . . . . 16  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
116115adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
117 ontri1 4607 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )  ->  ( x  C_  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  <->  -.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  x ) )
118113, 116, 117syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  -> 
( x  C_  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  <->  -.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  x ) )
119112, 118mpbird 224 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  x  C_  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
120 intss1 4057 . . . . . . . . . . . . . 14  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
121120adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
122119, 121eqssd 3357 . . . . . . . . . . . 12  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
12385, 122syldan 457 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  ( A `  x )  =/=  ( B `  x ) )  ->  x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
12479, 123sylan2 461 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) )  ->  x  =  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
125124fveq2d 5724 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) )  ->  ( A `  x )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
126124fveq2d 5724 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) )  ->  ( B `  x )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
127125, 126breq12d 4217 . . . . . . . 8  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) )  ->  (
( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x )  <->  ( A `  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
128127biimpd 199 . . . . . . 7  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) )  ->  (
( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x )  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) )
129128ex 424 . . . . . 6  |-  ( ( x  e.  On  /\  A. y  e.  x  ( A `  y )  =  ( B `  y ) )  -> 
( ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x )  ->  (
( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x )  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) ) )
130129pm2.43d 46 . . . . 5  |-  ( ( x  e.  On  /\  A. y  e.  x  ( A `  y )  =  ( B `  y ) )  -> 
( ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x )  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) )
131130expimpd 587 . . . 4  |-  ( x  e.  On  ->  (
( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) )  -> 
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
132131rexlimiv 2816 . . 3  |-  ( E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) )  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) )
13353, 132impbid1 195 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  <->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) ) )
1341, 133bitr4d 248 1  |-  ( ( 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 ) } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   (/)c0 3620   {ctp 3808   <.cop 3809   |^|cint 4042   class class class wbr 4204   Oncon0 4573   suc csuc 4575   ` cfv 5446   1oc1o 6709   2oc2o 6710   Nocsur 25587   < scslt 25588
This theorem is referenced by:  sltsgn1  25608  sltsgn2  25609  sltintdifex  25610  sltres  25611  nodenselem8  25635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-iota 5410  df-fv 5454  df-1o 6716  df-2o 6717  df-slt 25591
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