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Theorem sltval2 28993
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 28984 . 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 5874 . . . . . . . . . . . . 13  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
3 fvex 5874 . . . . . . . . . . . . 13  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
42, 3brtp 28755 . . . . . . . . . . . 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 7142 . . . . . . . . . . . . . . . . 17  |-  1o  =/=  (/)
6 df-ne 2664 . . . . . . . . . . . . . . . . 17  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
75, 6mpbi 208 . . . . . . . . . . . . . . . 16  |-  -.  1o  =  (/)
8 eqeq1 2471 . . . . . . . . . . . . . . . 16  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  (
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  <->  1o  =  (/) ) )
97, 8mtbiri 303 . . . . . . . . . . . . . . 15  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  -.  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
10 fvprc 5858 . . . . . . . . . . . . . . 15  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
119, 10nsyl2 127 . . . . . . . . . . . . . 14  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
1211adantr 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 )
1311adantr 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 )
14 2on0 7136 . . . . . . . . . . . . . . . . 17  |-  2o  =/=  (/)
15 df-ne 2664 . . . . . . . . . . . . . . . . 17  |-  ( 2o  =/=  (/)  <->  -.  2o  =  (/) )
1614, 15mpbi 208 . . . . . . . . . . . . . . . 16  |-  -.  2o  =  (/)
17 eqeq1 2471 . . . . . . . . . . . . . . . 16  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  (
( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  <->  2o  =  (/) ) )
1816, 17mtbiri 303 . . . . . . . . . . . . . . 15  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
19 fvprc 5858 . . . . . . . . . . . . . . 15  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
2018, 19nsyl2 127 . . . . . . . . . . . . . 14  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
2120adantl 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 )
2212, 13, 213jaoi 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 )
234, 22sylbi 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 )
24 onintrab 6614 . . . . . . . . . . 11  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  _V  <->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
2523, 24sylib 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 )
2625adantl 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 )
27 onelon 4903 . . . . . . . . . . . . . . 15  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  -> 
y  e.  On )
2827expcom 435 . . . . . . . . . . . . . 14  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  ->  y  e.  On ) )
2926, 28syl5 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 ) )
30 fveq2 5864 . . . . . . . . . . . . . . . 16  |-  ( a  =  y  ->  ( A `  a )  =  ( A `  y ) )
31 fveq2 5864 . . . . . . . . . . . . . . . 16  |-  ( a  =  y  ->  ( B `  a )  =  ( B `  y ) )
3230, 31neeq12d 2746 . . . . . . . . . . . . . . 15  |-  ( a  =  y  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  y )  =/=  ( B `  y )
) )
3332onnminsb 6617 . . . . . . . . . . . . . 14  |-  ( y  e.  On  ->  (
y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  -.  ( A `  y
)  =/=  ( B `
 y ) ) )
3433com12 31 . . . . . . . . . . . . 13  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  (
y  e.  On  ->  -.  ( A `  y
)  =/=  ( B `
 y ) ) )
3529, 34syld 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 ) ) )
3635com12 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 )
) )
37 df-ne 2664 . . . . . . . . . . . 12  |-  ( ( A `  y )  =/=  ( B `  y )  <->  -.  ( A `  y )  =  ( B `  y ) )
3837con2bii 332 . . . . . . . . . . 11  |-  ( ( A `  y )  =  ( B `  y )  <->  -.  ( A `  y )  =/=  ( B `  y
) )
3936, 38syl6ibr 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 ) ) )
4039ralrimiv 2876 . . . . . . . . 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 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 ) ) )
4241ex 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 ) ) ) )
4342impac 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 ) } ) ) )
44 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 ) } ) ) ) )
4543, 44sylib 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 ) } ) ) ) )
46 raleq 3058 . . . . . . 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 5864 . . . . . . . 8  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  x )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
48 fveq2 5864 . . . . . . . 8  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  x )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
4947, 48breq12d 4460 . . . . . . 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 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 ) } ) ) ) )
5150rspcev 3214 . . . . 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 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 ) ) ) )
54 eqeq12 2486 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  (/) )  ->  (
( A `  x
)  =  ( B `
 x )  <->  1o  =  (/) ) )
557, 54mtbiri 303 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  (/) )  ->  -.  ( A `  x )  =  ( B `  x ) )
56 1on 7134 . . . . . . . . . . . . . . . . 17  |-  1o  e.  On
57 0elon 4931 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
58 suc11 4981 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =  suc  (/)  <->  1o  =  (/) ) )
5958necon3bid 2725 . . . . . . . . . . . . . . . . 17  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =/=  suc  (/) 
<->  1o  =/=  (/) ) )
6056, 57, 59mp2an 672 . . . . . . . . . . . . . . . 16  |-  ( suc 
1o  =/=  suc  (/)  <->  1o  =/=  (/) )
615, 60mpbir 209 . . . . . . . . . . . . . . 15  |-  suc  1o  =/=  suc  (/)
62 df-2o 7128 . . . . . . . . . . . . . . . 16  |-  2o  =  suc  1o
63 df-1o 7127 . . . . . . . . . . . . . . . 16  |-  1o  =  suc  (/)
6462, 63eqeq12i 2487 . . . . . . . . . . . . . . 15  |-  ( 2o  =  1o  <->  suc  1o  =  suc  (/) )
6561, 64nemtbir 2795 . . . . . . . . . . . . . 14  |-  -.  2o  =  1o
66 eqeq12 2486 . . . . . . . . . . . . . . 15  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  -> 
( ( A `  x )  =  ( B `  x )  <-> 
1o  =  2o ) )
67 eqcom 2476 . . . . . . . . . . . . . . 15  |-  ( 1o  =  2o  <->  2o  =  1o )
6866, 67syl6bb 261 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  -> 
( ( A `  x )  =  ( B `  x )  <-> 
2o  =  1o ) )
6965, 68mtbiri 303 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  ->  -.  ( A `  x
)  =  ( B `
 x ) )
70 eqcom 2476 . . . . . . . . . . . . . . 15  |-  ( 2o  =  (/)  <->  (/)  =  2o )
7116, 70mtbi 298 . . . . . . . . . . . . . 14  |-  -.  (/)  =  2o
72 eqeq12 2486 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  (/)  /\  ( B `  x )  =  2o )  ->  (
( A `  x
)  =  ( B `
 x )  <->  (/)  =  2o ) )
7371, 72mtbiri 303 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  (/)  /\  ( B `  x )  =  2o )  ->  -.  ( A `  x )  =  ( B `  x ) )
7455, 69, 733jaoi 1291 . . . . . . . . . . . 12  |-  ( ( ( ( A `  x )  =  1o 
/\  ( B `  x )  =  (/) )  \/  ( ( A `  x )  =  1o  /\  ( B `  x )  =  2o )  \/  (
( A `  x
)  =  (/)  /\  ( B `  x )  =  2o ) )  ->  -.  ( A `  x
)  =  ( B `
 x ) )
75 fvex 5874 . . . . . . . . . . . . 13  |-  ( A `
 x )  e. 
_V
76 fvex 5874 . . . . . . . . . . . . 13  |-  ( B `
 x )  e. 
_V
7775, 76brtp 28755 . . . . . . . . . . . 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 2664 . . . . . . . . . . . 12  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
7974, 77, 783imtr4i 266 . . . . . . . . . . 11  |-  ( ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
)  ->  ( A `  x )  =/=  ( B `  x )
)
80 fveq2 5864 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( A `  a )  =  ( A `  x ) )
81 fveq2 5864 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( B `  a )  =  ( B `  x ) )
8280, 81neeq12d 2746 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  x )  =/=  ( B `  x )
) )
8382elrab 3261 . . . . . . . . . . . . . 14  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  <->  ( x  e.  On  /\  ( A `
 x )  =/=  ( B `  x
) ) )
8483biimpri 206 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  ( A `  x )  =/=  ( B `  x ) )  ->  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
8584adantlr 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
) } )
86 ssrab2 3585 . . . . . . . . . . . . . . . . . 18  |-  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On
87 ne0i 3791 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  =/=  (/) )
8887adantl 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 ) }  =/=  (/) )
89 onint 6608 . . . . . . . . . . . . . . . . . 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 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
) } )
91 nfrab1 3042 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
9291nfint 4292 . . . . . . . . . . . . . . . . . . 19  |-  F/_ a |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
93 nfcv 2629 . . . . . . . . . . . . . . . . . . 19  |-  F/_ a On
94 nfcv 2629 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ a A
9594, 92nffv 5871 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
96 nfcv 2629 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ a B
9796, 92nffv 5871 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a
( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
9895, 97nfne 2798 . . . . . . . . . . . . . . . . . . 19  |-  F/ a ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =/=  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
99 fveq2 5864 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  a )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
100 fveq2 5864 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  a )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
10199, 100neeq12d 2746 . . . . . . . . . . . . . . . . . . 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 3259 . . . . . . . . . . . . . . . . . 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 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 ) } ) )
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 2664 . . . . . . . . . . . . . . . 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 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
) } ) )
107 fveq2 5864 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  y )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
108 fveq2 5864 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  y )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
109107, 108eqeq12d 2489 . . . . . . . . . . . . . . . . 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 3211 . . . . . . . . . . . . . . . 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 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
) } ) ) )
112106, 111mtod 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
)
113 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 )
114 oninton 6613 . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . 16  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
116115adantl 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 )
117 ontri1 4912 . . . . . . . . . . . . . . 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 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 ) )
119112, 118mpbird 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
) } )
120 intss1 4297 . . . . . . . . . . . . . 14  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
121120adantl 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 )
122119, 121eqssd 3521 . . . . . . . . . . . 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 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 ) } )
12479, 123sylan2 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
) } )
125124fveq2d 5868 . . . . . . . . 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 5868 . . . . . . . . 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 4460 . . . . . . . 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 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 ) } ) ) )
129128ex 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 ) } ) ) ) )
130129pm2.43d 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 ) } ) ) )
131130expimpd 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 ) } ) ) )
132131rexlimiv 2949 . . 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 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 ) ) ) )
1341, 133bitr4d 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {ctp 4031   <.cop 4033   |^|cint 4282   class class class wbr 4447   Oncon0 4878   suc csuc 4880   ` cfv 5586   1oc1o 7120   2oc2o 7121   Nocsur 28977   <scslt 28978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-iota 5549  df-fv 5594  df-1o 7127  df-2o 7128  df-slt 28981
This theorem is referenced by:  sltsgn1  28998  sltsgn2  28999  sltintdifex  29000  sltres  29001  nodenselem8  29025
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