Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sltval2 Unicode version

Theorem sltval2 24312
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 24303 . 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 5541 . . . . . . . . . . . . 13  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
3 fvex 5541 . . . . . . . . . . . . 13  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
42, 3brtp 24108 . . . . . . . . . . . 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 6496 . . . . . . . . . . . . . . . . 17  |-  1o  =/=  (/)
6 df-ne 2450 . . . . . . . . . . . . . . . . 17  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
75, 6mpbi 199 . . . . . . . . . . . . . . . 16  |-  -.  1o  =  (/)
8 eqeq1 2291 . . . . . . . . . . . . . . . 16  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  (
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  <->  1o  =  (/) ) )
97, 8mtbiri 294 . . . . . . . . . . . . . . 15  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  -.  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
10 fvprc 5521 . . . . . . . . . . . . . . 15  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
119, 10nsyl2 119 . . . . . . . . . . . . . 14  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
1211adantr 451 . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 6490 . . . . . . . . . . . . . . . . 17  |-  2o  =/=  (/)
15 df-ne 2450 . . . . . . . . . . . . . . . . 17  |-  ( 2o  =/=  (/)  <->  -.  2o  =  (/) )
1614, 15mpbi 199 . . . . . . . . . . . . . . . 16  |-  -.  2o  =  (/)
17 eqeq1 2291 . . . . . . . . . . . . . . . 16  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  (
( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  <->  2o  =  (/) ) )
1816, 17mtbiri 294 . . . . . . . . . . . . . . 15  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
19 fvprc 5521 . . . . . . . . . . . . . . 15  |-  ( -. 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V  ->  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
2018, 19nsyl2 119 . . . . . . . . . . . . . 14  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  _V )
2120adantl 452 . . . . . . . . . . . . 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 1245 . . . . . . . . . . . 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 187 . . . . . . . . . . 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 4594 . . . . . . . . . . 11  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  _V  <->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
2523, 24sylib 188 . . . . . . . . . 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 452 . . . . . . . . 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 4419 . . . . . . . . . . . . . . 15  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  -> 
y  e.  On )
2827expcom 424 . . . . . . . . . . . . . 14  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  ->  y  e.  On ) )
2926, 28syl5 28 . . . . . . . . . . . . 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 5527 . . . . . . . . . . . . . . . 16  |-  ( a  =  y  ->  ( A `  a )  =  ( A `  y ) )
31 fveq2 5527 . . . . . . . . . . . . . . . 16  |-  ( a  =  y  ->  ( B `  a )  =  ( B `  y ) )
3230, 31neeq12d 2463 . . . . . . . . . . . . . . 15  |-  ( a  =  y  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  y )  =/=  ( B `  y )
) )
3332onnminsb 4597 . . . . . . . . . . . . . 14  |-  ( y  e.  On  ->  (
y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  -.  ( A `  y
)  =/=  ( B `
 y ) ) )
3433com12 27 . . . . . . . . . . . . 13  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  (
y  e.  On  ->  -.  ( A `  y
)  =/=  ( B `
 y ) ) )
3529, 34syld 40 . . . . . . . . . . . 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 27 . . . . . . . . . . 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 2450 . . . . . . . . . . . 12  |-  ( ( A `  y )  =/=  ( B `  y )  <->  -.  ( A `  y )  =  ( B `  y ) )
3837con2bii 322 . . . . . . . . . . 11  |-  ( ( A `  y )  =  ( B `  y )  <->  -.  ( A `  y )  =/=  ( B `  y
) )
3936, 38syl6ibr 218 . . . . . . . . . 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 2627 . . . . . . . . 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 518 . . . . . . . 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 423 . . . . . . 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 604 . . . . . 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 630 . . . . . 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 188 . . . . 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 2738 . . . . . . 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 5527 . . . . . . . 8  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  x )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
48 fveq2 5527 . . . . . . . 8  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  x )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
4947, 48breq12d 4038 . . . . . . 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 691 . . . . . 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 2886 . . . . 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 15 . . . 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 423 . . 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 2297 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  (/) )  ->  (
( A `  x
)  =  ( B `
 x )  <->  1o  =  (/) ) )
557, 54mtbiri 294 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  (/) )  ->  -.  ( A `  x )  =  ( B `  x ) )
56 1on 6488 . . . . . . . . . . . . . . . . 17  |-  1o  e.  On
57 0elon 4447 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
58 suc11 4498 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =  suc  (/)  <->  1o  =  (/) ) )
5958necon3bid 2483 . . . . . . . . . . . . . . . . 17  |-  ( ( 1o  e.  On  /\  (/) 
e.  On )  -> 
( suc  1o  =/=  suc  (/) 
<->  1o  =/=  (/) ) )
6056, 57, 59mp2an 653 . . . . . . . . . . . . . . . 16  |-  ( suc 
1o  =/=  suc  (/)  <->  1o  =/=  (/) )
615, 60mpbir 200 . . . . . . . . . . . . . . 15  |-  suc  1o  =/=  suc  (/)
62 df-2o 6482 . . . . . . . . . . . . . . . 16  |-  2o  =  suc  1o
63 df-1o 6481 . . . . . . . . . . . . . . . 16  |-  1o  =  suc  (/)
6462, 63eqeq12i 2298 . . . . . . . . . . . . . . 15  |-  ( 2o  =  1o  <->  suc  1o  =  suc  (/) )
6561, 64nemtbir 2536 . . . . . . . . . . . . . 14  |-  -.  2o  =  1o
66 eqeq12 2297 . . . . . . . . . . . . . . 15  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  -> 
( ( A `  x )  =  ( B `  x )  <-> 
1o  =  2o ) )
67 eqcom 2287 . . . . . . . . . . . . . . 15  |-  ( 1o  =  2o  <->  2o  =  1o )
6866, 67syl6bb 252 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  -> 
( ( A `  x )  =  ( B `  x )  <-> 
2o  =  1o ) )
6965, 68mtbiri 294 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  1o  /\  ( B `  x )  =  2o )  ->  -.  ( A `  x
)  =  ( B `
 x ) )
70 eqcom 2287 . . . . . . . . . . . . . . 15  |-  ( 2o  =  (/)  <->  (/)  =  2o )
7116, 70mtbi 289 . . . . . . . . . . . . . 14  |-  -.  (/)  =  2o
72 eqeq12 2297 . . . . . . . . . . . . . 14  |-  ( ( ( A `  x
)  =  (/)  /\  ( B `  x )  =  2o )  ->  (
( A `  x
)  =  ( B `
 x )  <->  (/)  =  2o ) )
7371, 72mtbiri 294 . . . . . . . . . . . . 13  |-  ( ( ( A `  x
)  =  (/)  /\  ( B `  x )  =  2o )  ->  -.  ( A `  x )  =  ( B `  x ) )
7455, 69, 733jaoi 1245 . . . . . . . . . . . 12  |-  ( ( ( ( A `  x )  =  1o 
/\  ( B `  x )  =  (/) )  \/  ( ( A `  x )  =  1o  /\  ( B `  x )  =  2o )  \/  (
( A `  x
)  =  (/)  /\  ( B `  x )  =  2o ) )  ->  -.  ( A `  x
)  =  ( B `
 x ) )
75 fvex 5541 . . . . . . . . . . . . 13  |-  ( A `
 x )  e. 
_V
76 fvex 5541 . . . . . . . . . . . . 13  |-  ( B `
 x )  e. 
_V
7775, 76brtp 24108 . . . . . . . . . . . 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 2450 . . . . . . . . . . . 12  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
7974, 77, 783imtr4i 257 . . . . . . . . . . 11  |-  ( ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
)  ->  ( A `  x )  =/=  ( B `  x )
)
80 fveq2 5527 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( A `  a )  =  ( A `  x ) )
81 fveq2 5527 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( B `  a )  =  ( B `  x ) )
8280, 81neeq12d 2463 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  x )  =/=  ( B `  x )
) )
8382elrab 2925 . . . . . . . . . . . . . 14  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  <->  ( x  e.  On  /\  ( A `
 x )  =/=  ( B `  x
) ) )
8483biimpri 197 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  ( A `  x )  =/=  ( B `  x ) )  ->  x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
8584adantlr 695 . . . . . . . . . . . 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 3260 . . . . . . . . . . . . . . . . . 18  |-  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  On
87 ne0i 3463 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  =/=  (/) )
8887adantl 452 . . . . . . . . . . . . . . . . . 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 4588 . . . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . . 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 2722 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
9291nfint 3874 . . . . . . . . . . . . . . . . . . 19  |-  F/_ a |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }
93 nfcv 2421 . . . . . . . . . . . . . . . . . . 19  |-  F/_ a On
94 nfcv 2421 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ a A
9594, 92nffv 5534 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
96 nfcv 2421 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ a B
9796, 92nffv 5534 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ a
( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )
9895, 97nfne 2541 . . . . . . . . . . . . . . . . . . 19  |-  F/ a ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =/=  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
99 fveq2 5527 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  a )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
100 fveq2 5527 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  a )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
10199, 100neeq12d 2463 . . . . . . . . . . . . . . . . . . 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 2924 . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . . 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 188 . . . . . . . . . . . . . . 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 5527 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  y )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
108 fveq2 5527 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  y )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
109107, 108eqeq12d 2299 . . . . . . . . . . . . . . . . 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 2883 . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . 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 168 . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 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 4593 . . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . 16  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
116115adantl 452 . . . . . . . . . . . . . . 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 4428 . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 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 3879 . . . . . . . . . . . . . 14  |-  ( x  e.  { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
121120adantl 452 . . . . . . . . . . . . 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 3198 . . . . . . . . . . . 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 456 . . . . . . . . . . 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 460 . . . . . . . . . 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 5531 . . . . . . . . 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 5531 . . . . . . . . 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 4038 . . . . . . . 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 198 . . . . . . 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 423 . . . . . 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 44 . . . . 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 586 . . . 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 2663 . . 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 194 . 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 247 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 176    /\ wa 358    \/ w3o 933    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   {crab 2549   _Vcvv 2790    C_ wss 3154   (/)c0 3457   {ctp 3644   <.cop 3645   |^|cint 3864   class class class wbr 4025   Oncon0 4394   suc csuc 4396   ` cfv 5257   1oc1o 6474   2oc2o 6475   Nocsur 24296   < scslt 24297
This theorem is referenced by:  sltsgn1  24317  sltsgn2  24318  sltintdifex  24319  sltres  24320  nodenselem8  24344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-suc 4400  df-iota 5221  df-fv 5265  df-1o 6481  df-2o 6482  df-slt 24300
  Copyright terms: Public domain W3C validator