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Theorem sltval2 29590
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 29581 . 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 5882 . . . . . . . . . . . . 13  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
3 fvex 5882 . . . . . . . . . . . . 13  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
42, 3brtp 29353 . . . . . . . . . . . 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 7163 . . . . . . . . . . . . . . . . 17  |-  1o  =/=  (/)
65neii 2656 . . . . . . . . . . . . . . . 16  |-  -.  1o  =  (/)
7 eqeq1 2461 . . . . . . . . . . . . . . . 16  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  (
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  <->  1o  =  (/) ) )
86, 7mtbiri 303 . . . . . . . . . . . . . . 15  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  -.  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
9 fvprc 5866 . . . . . . . . . . . . . . 15  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
108, 9nsyl2 127 . . . . . . . . . . . . . 14  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
1110adantr 465 . . . . . . . . . . . . 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 )
1210adantr 465 . . . . . . . . . . . . 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 )
13 2on0 7157 . . . . . . . . . . . . . . . . 17  |-  2o  =/=  (/)
1413neii 2656 . . . . . . . . . . . . . . . 16  |-  -.  2o  =  (/)
15 eqeq1 2461 . . . . . . . . . . . . . . . 16  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  (
( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  <->  2o  =  (/) ) )
1614, 15mtbiri 303 . . . . . . . . . . . . . . 15  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
17 fvprc 5866 . . . . . . . . . . . . . . 15  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
1816, 17nsyl2 127 . . . . . . . . . . . . . 14  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
1918adantl 466 . . . . . . . . . . . . 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 )
2011, 12, 193jaoi 1291 . . . . . . . . . . . 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 )
214, 20sylbi 195 . . . . . . . . . . 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 )
22 onintrab 6635 . . . . . . . . . . 11  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  _V  <->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
2321, 22sylib 196 . . . . . . . . . 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 )
2423adantl 466 . . . . . . . . 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 )
25 onelon 4912 . . . . . . . . . . . . . . 15  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  -> 
y  e.  On )
2625expcom 435 . . . . . . . . . . . . . 14  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  ->  y  e.  On ) )
2724, 26syl5 32 . . . . . . . . . . . . 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 ) )
28 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( a  =  y  ->  ( A `  a )  =  ( A `  y ) )
29 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( a  =  y  ->  ( B `  a )  =  ( B `  y ) )
3028, 29neeq12d 2736 . . . . . . . . . . . . . . 15  |-  ( a  =  y  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  y )  =/=  ( B `  y )
) )
3130onnminsb 6638 . . . . . . . . . . . . . 14  |-  ( y  e.  On  ->  (
y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  -.  ( A `  y
)  =/=  ( B `
 y ) ) )
3231com12 31 . . . . . . . . . . . . 13  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  (
y  e.  On  ->  -.  ( A `  y
)  =/=  ( B `
 y ) ) )
3327, 32syld 44 . . . . . . . . . . . 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 ) ) )
3433com12 31 . . . . . . . . . . 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 )
) )
35 df-ne 2654 . . . . . . . . . . . 12  |-  ( ( A `  y )  =/=  ( B `  y )  <->  -.  ( A `  y )  =  ( B `  y ) )
3635con2bii 332 . . . . . . . . . . 11  |-  ( ( A `  y )  =  ( B `  y )  <->  -.  ( A `  y )  =/=  ( B `  y
) )
3734, 36syl6ibr 227 . . . . . . . . . 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 ) ) )
3837ralrimiv 2869 . . . . . . . . 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 ) )
3924, 38jca 532 . . . . . . . 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 ) ) )
4039ex 434 . . . . . . 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 ) ) ) )
4140impac 621 . . . . . 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 ) } ) ) )
42 anass 649 . . . . . 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 ) } ) ) ) )
4341, 42sylib 196 . . . . 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 ) } ) ) ) )
44 raleq 3054 . . . . . . 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
) ) )
45 fveq2 5872 . . . . . . . 8  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  x )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
46 fveq2 5872 . . . . . . . 8  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  x )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
4745, 46breq12d 4469 . . . . . . 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 ) } ) ) )
4844, 47anbi12d 710 . . . . . 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 ) } ) ) ) )
4948rspcev 3210 . . . . 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
) ) )
5043, 49syl 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
) ) )
5150ex 434 . . 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 ) ) ) )
52 eqeq12 2476 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  (/) )  ->  (
( A `  x
)  =  ( B `
 x )  <->  1o  =  (/) ) )
536, 52mtbiri 303 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  (/) )  ->  -.  ( A `  x )  =  ( B `  x ) )
54 1on 7155 . . . . . . . . . . . . . . . . 17  |-  1o  e.  On
55 0elon 4940 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
56 suc11 4990 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =  suc  (/)  <->  1o  =  (/) ) )
5756necon3bid 2715 . . . . . . . . . . . . . . . . 17  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =/=  suc  (/) 
<->  1o  =/=  (/) ) )
5854, 55, 57mp2an 672 . . . . . . . . . . . . . . . 16  |-  ( suc 
1o  =/=  suc  (/)  <->  1o  =/=  (/) )
595, 58mpbir 209 . . . . . . . . . . . . . . 15  |-  suc  1o  =/=  suc  (/)
60 df-2o 7149 . . . . . . . . . . . . . . . 16  |-  2o  =  suc  1o
61 df-1o 7148 . . . . . . . . . . . . . . . 16  |-  1o  =  suc  (/)
6260, 61eqeq12i 2477 . . . . . . . . . . . . . . 15  |-  ( 2o  =  1o  <->  suc  1o  =  suc  (/) )
6359, 62nemtbir 2785 . . . . . . . . . . . . . 14  |-  -.  2o  =  1o
64 eqeq12 2476 . . . . . . . . . . . . . . 15  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  -> 
( ( A `  x )  =  ( B `  x )  <-> 
1o  =  2o ) )
65 eqcom 2466 . . . . . . . . . . . . . . 15  |-  ( 1o  =  2o  <->  2o  =  1o )
6664, 65syl6bb 261 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  -> 
( ( A `  x )  =  ( B `  x )  <-> 
2o  =  1o ) )
6763, 66mtbiri 303 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  ->  -.  ( A `  x
)  =  ( B `
 x ) )
6813nesymi 2730 . . . . . . . . . . . . . 14  |-  -.  (/)  =  2o
69 eqeq12 2476 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  (/)  /\  ( B `  x )  =  2o )  ->  (
( A `  x
)  =  ( B `
 x )  <->  (/)  =  2o ) )
7068, 69mtbiri 303 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  (/)  /\  ( B `  x )  =  2o )  ->  -.  ( A `  x )  =  ( B `  x ) )
7153, 67, 703jaoi 1291 . . . . . . . . . . . 12  |-  ( ( ( ( A `  x )  =  1o 
/\  ( B `  x )  =  (/) )  \/  ( ( A `  x )  =  1o  /\  ( B `  x )  =  2o )  \/  (
( A `  x
)  =  (/)  /\  ( B `  x )  =  2o ) )  ->  -.  ( A `  x
)  =  ( B `
 x ) )
72 fvex 5882 . . . . . . . . . . . . 13  |-  ( A `
 x )  e. 
_V
73 fvex 5882 . . . . . . . . . . . . 13  |-  ( B `
 x )  e. 
_V
7472, 73brtp 29353 . . . . . . . . . . . 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 ) ) )
75 df-ne 2654 . . . . . . . . . . . 12  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
7671, 74, 753imtr4i 266 . . . . . . . . . . 11  |-  ( ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
)  ->  ( A `  x )  =/=  ( B `  x )
)
77 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( A `  a )  =  ( A `  x ) )
78 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( B `  a )  =  ( B `  x ) )
7977, 78neeq12d 2736 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  x )  =/=  ( B `  x )
) )
8079elrab 3257 . . . . . . . . . . . . . 14  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  <->  ( x  e.  On  /\  ( A `
 x )  =/=  ( B `  x
) ) )
8180biimpri 206 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  ( A `  x )  =/=  ( B `  x ) )  ->  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
8281adantlr 714 . . . . . . . . . . . 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
) } )
83 ssrab2 3581 . . . . . . . . . . . . . . . . . 18  |-  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On
84 ne0i 3799 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  =/=  (/) )
8584adantl 466 . . . . . . . . . . . . . . . . . 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 ) }  =/=  (/) )
86 onint 6629 . . . . . . . . . . . . . . . . . 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 ) } )
8783, 85, 86sylancr 663 . . . . . . . . . . . . . . . . 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
) } )
88 nfrab1 3038 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
8988nfint 4298 . . . . . . . . . . . . . . . . . . 19  |-  F/_ a |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
90 nfcv 2619 . . . . . . . . . . . . . . . . . . 19  |-  F/_ a On
91 nfcv 2619 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ a A
9291, 89nffv 5879 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
93 nfcv 2619 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ a B
9493, 89nffv 5879 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a
( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
9592, 94nfne 2788 . . . . . . . . . . . . . . . . . . 19  |-  F/ a ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =/=  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
96 fveq2 5872 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  a )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
97 fveq2 5872 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  a )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
9896, 97neeq12d 2736 . . . . . . . . . . . . . . . . . . 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 ) } ) ) )
9989, 90, 95, 98elrabf 3255 . . . . . . . . . . . . . . . . . 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 ) } ) ) )
10099simprbi 464 . . . . . . . . . . . . . . . . 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 ) } ) )
10187, 100syl 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
) } ) )
102 df-ne 2654 . . . . . . . . . . . . . . . 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 ) } ) )
103101, 102sylib 196 . . . . . . . . . . . . . . 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
) } ) )
104 fveq2 5872 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  y )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
105 fveq2 5872 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  y )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
106104, 105eqeq12d 2479 . . . . . . . . . . . . . . . . 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 ) } ) ) )
107106rspccv 3207 . . . . . . . . . . . . . . . 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 ) } ) ) )
108107ad2antlr 726 . . . . . . . . . . . . . . 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
) } ) ) )
109103, 108mtod 177 . . . . . . . . . . . . . 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
)
110 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  ->  x  e.  On )
111 oninton 6634 . . . . . . . . . . . . . . . . 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 )
11283, 84, 111sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
113112adantl 466 . . . . . . . . . . . . . . 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 )
114 ontri1 4921 . . . . . . . . . . . . . . 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 ) )
115110, 113, 114syl2anc 661 . . . . . . . . . . . . . 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 ) )
116109, 115mpbird 232 . . . . . . . . . . . . 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
) } )
117 intss1 4303 . . . . . . . . . . . . . 14  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
118117adantl 466 . . . . . . . . . . . . 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 )
119116, 118eqssd 3516 . . . . . . . . . . . 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 ) } )
12082, 119syldan 470 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\ 
A. y  e.  x  ( A `  y )  =  ( B `  y ) )  /\  ( A `  x )  =/=  ( B `  x ) )  ->  x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
12176, 120sylan2 474 . . . . . . . . . 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
) } )
122121fveq2d 5876 . . . . . . . . 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 ) } ) )
123121fveq2d 5876 . . . . . . . . 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 ) } ) )
124122, 123breq12d 4469 . . . . . . . 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 ) } ) ) )
125124biimpd 207 . . . . . . 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 ) } ) ) )
126125ex 434 . . . . . 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 ) } ) ) ) )
127126pm2.43d 48 . . . . 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 ) } ) ) )
128127expimpd 603 . . . 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 ) } ) ) )
129128rexlimiv 2943 . . 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 ) } ) )
13051, 129impbid1 203 . 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 ) ) ) )
1311, 130bitr4d 256 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   {ctp 4036   <.cop 4038   |^|cint 4288   class class class wbr 4456   Oncon0 4887   suc csuc 4889   ` cfv 5594   1oc1o 7141   2oc2o 7142   Nocsur 29574   <scslt 29575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-iota 5557  df-fv 5602  df-1o 7148  df-2o 7149  df-slt 29578
This theorem is referenced by:  sltsgn1  29595  sltsgn2  29596  sltintdifex  29597  sltres  29598  nodenselem8  29622
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