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Theorem nodenselem8 29025
Description: Lemma for nodense 29026. Give a condition for surreal less than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)
Assertion
Ref Expression
nodenselem8  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  ( A <s B  <->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) )
Distinct variable groups:    A, a    B, a

Proof of Theorem nodenselem8
StepHypRef Expression
1 nodenselem5 29022 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )
)
21exp32 605 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  =  ( bday `  B )  ->  ( A <s B  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A ) ) ) )
323impia 1193 . . 3  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  ( A <s B  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A ) ) )
4 sltval2 28993 . . . . 5  |-  ( ( 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 ) } ) ) )
543adant3 1016 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  ( A <s B  <->  ( A `  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
6 fvex 5874 . . . . . 6  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
7 fvex 5874 . . . . . 6  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
86, 7brtp 28755 . . . . 5  |-  ( ( 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 ) ) )
9 eleq2 2540 . . . . . . . . . . . . 13  |-  ( (
bday `  A )  =  ( bday `  B
)  ->  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
)  <->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  (
bday `  B )
) )
109biimpd 207 . . . . . . . . . . . 12  |-  ( (
bday `  A )  =  ( bday `  B
)  ->  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
)  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B )
) )
11 nosgnn0 28995 . . . . . . . . . . . . . . 15  |-  -.  (/)  e.  { 1o ,  2o }
12 nofnbday 28989 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  No  ->  B  Fn  ( bday `  B
) )
13 fnfvelrn 6016 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  Fn  ( bday `  B )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  B
) )  ->  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  ran  B )
14 eleq1 2539 . . . . . . . . . . . . . . . . . 18  |-  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  ran  B  <->  (/)  e.  ran  B ) )
1513, 14syl5ibcom 220 . . . . . . . . . . . . . . . . 17  |-  ( ( B  Fn  ( bday `  B )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  B
) )  ->  (
( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  ->  (/)  e.  ran  B ) )
1612, 15sylan 471 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B ) )  -> 
( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  -> 
(/)  e.  ran  B ) )
17 norn 28988 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  No  ->  ran  B 
C_  { 1o ,  2o } )
1817sseld 3503 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  No  ->  ( (/) 
e.  ran  B  ->  (/)  e.  { 1o ,  2o } ) )
1918adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B ) )  -> 
( (/)  e.  ran  B  -> 
(/)  e.  { 1o ,  2o } ) )
2016, 19syld 44 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B ) )  -> 
( ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  -> 
(/)  e.  { 1o ,  2o } ) )
2111, 20mtoi 178 . . . . . . . . . . . . . 14  |-  ( ( B  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B ) )  ->  -.  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/) )
2221ex 434 . . . . . . . . . . . . 13  |-  ( B  e.  No  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B )  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) ) )
2322adantl 466 . . . . . . . . . . . 12  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B )  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) ) )
2410, 23syl9r 72 . . . . . . . . . . 11  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  =  ( bday `  B )  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) ) ) )
25243impia 1193 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) ) )
2625imp 429 . . . . . . . . 9  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
) )  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
2726intnand 914 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
) )  ->  -.  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o 
/\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) ) )
28 nofnbday 28989 . . . . . . . . . . . . 13  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
29 fnfvelrn 6016 . . . . . . . . . . . . . 14  |-  ( ( A  Fn  ( bday `  A )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
) )  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  ran  A )
30 eleq1 2539 . . . . . . . . . . . . . 14  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  ran  A  <->  (/)  e.  ran  A ) )
3129, 30syl5ibcom 220 . . . . . . . . . . . . 13  |-  ( ( A  Fn  ( bday `  A )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
) )  ->  (
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/)  ->  (/)  e.  ran  A ) )
3228, 31sylan 471 . . . . . . . . . . . 12  |-  ( ( A  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A ) )  -> 
( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  -> 
(/)  e.  ran  A ) )
33 norn 28988 . . . . . . . . . . . . . 14  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
3433sseld 3503 . . . . . . . . . . . . 13  |-  ( A  e.  No  ->  ( (/) 
e.  ran  A  ->  (/)  e.  { 1o ,  2o } ) )
3534adantr 465 . . . . . . . . . . . 12  |-  ( ( A  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A ) )  -> 
( (/)  e.  ran  A  -> 
(/)  e.  { 1o ,  2o } ) )
3632, 35syld 44 . . . . . . . . . . 11  |-  ( ( A  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A ) )  -> 
( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  -> 
(/)  e.  { 1o ,  2o } ) )
3711, 36mtoi 178 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A ) )  ->  -.  ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/) )
38373ad2antl1 1158 . . . . . . . . 9  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
) )  ->  -.  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )
3938intnanrd 915 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
) )  ->  -.  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  /\  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  2o ) )
40 3orel13 28569 . . . . . . . 8  |-  ( ( -.  ( ( 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 ) } )  =  (/)  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) )  ->  ( ( ( ( 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 `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) )
4127, 39, 40syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  /\  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  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 ) )  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) )
4241ex 434 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  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 ) )  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) ) )
4342com23 78 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  (
( ( ( 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.  ( bday `  A
)  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) ) )
448, 43syl5bi 217 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  (
( 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.  ( bday `  A )  ->  (
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) ) )
455, 44sylbid 215 . . 3  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  ( A <s B  -> 
( |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  (
bday `  A )  ->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o 
/\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) ) )
463, 45mpdd 40 . 2  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  ( A <s B  -> 
( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o 
/\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) )
47 3mix2 1166 . . . 4  |-  ( ( ( 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
) } )  =  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 ) ) )
4847, 8sylibr 212 . . 3  |-  ( ( ( 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 ) } ) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
4948, 5syl5ibr 221 . 2  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  (
( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o 
/\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )  ->  A <s B ) )
5046, 49impbid 191 1  |-  ( ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B
) )  ->  ( A <s B  <->  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818   (/)c0 3785   {cpr 4029   {ctp 4031   <.cop 4033   |^|cint 4282   class class class wbr 4447   Oncon0 4878   ran crn 5000    Fn wfn 5581   ` cfv 5586   1oc1o 7120   2oc2o 7121   Nocsur 28977   <scslt 28978   bdaycbday 28979
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-rep 4558  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-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  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-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-1o 7127  df-2o 7128  df-no 28980  df-slt 28981  df-bday 28982
This theorem is referenced by:  nodense  29026
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