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Theorem nodenselem8 27780
Description: Lemma for nodense 27781. 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 27777 . . . . 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 1184 . . 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 27748 . . . . 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 1008 . . . 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 5696 . . . . . 6  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
7 fvex 5696 . . . . . 6  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
86, 7brtp 27510 . . . . 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 2499 . . . . . . . . . . . . 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 27750 . . . . . . . . . . . . . . 15  |-  -.  (/)  e.  { 1o ,  2o }
12 nofnbday 27744 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  No  ->  B  Fn  ( bday `  B
) )
13 fnfvelrn 5835 . . . . . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . . . . . . 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 27743 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  No  ->  ran  B 
C_  { 1o ,  2o } )
1817sseld 3350 . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . 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 907 . . . . . . . 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 27744 . . . . . . . . . . . . 13  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
29 fnfvelrn 5835 . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . . 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 27743 . . . . . . . . . . . . . 14  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
3433sseld 3350 . . . . . . . . . . . . 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 1150 . . . . . . . . 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 908 . . . . . . . 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 27324 . . . . . . . 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 1158 . . . 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 964    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   {crab 2714   (/)c0 3632   {cpr 3874   {ctp 3876   <.cop 3878   |^|cint 4123   class class class wbr 4287   Oncon0 4714   ran crn 4836    Fn wfn 5408   ` cfv 5413   1oc1o 6905   2oc2o 6906   Nocsur 27732   <scslt 27733   bdaycbday 27734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6912  df-2o 6913  df-no 27735  df-slt 27736  df-bday 27737
This theorem is referenced by:  nodense  27781
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