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Theorem nodenselem8 30353
Description: Lemma for nodense 30354. 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 30350 . . . . 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 608 . . . 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 1202 . . 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 30321 . . . . 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 1025 . . . 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 5891 . . . . . 6  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
7 fvex 5891 . . . . . 6  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
86, 7brtp 30167 . . . . 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 2502 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 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 30323 . . . . . . . . . . . . . . 15  |-  -.  (/)  e.  { 1o ,  2o }
12 nofnbday 30317 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  No  ->  B  Fn  ( bday `  B
) )
13 fnfvelrn 6034 . . . . . . . . . . . . . . . . . 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 2501 . . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . 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 30316 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  No  ->  ran  B 
C_  { 1o ,  2o } )
1817sseld 3469 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  No  ->  ( (/) 
e.  ran  B  ->  (/)  e.  { 1o ,  2o } ) )
1918adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B ) )  -> 
( (/)  e.  ran  B  -> 
(/)  e.  { 1o ,  2o } ) )
2016, 19syld 45 . . . . . . . . . . . . . . 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 181 . . . . . . . . . . . . . 14  |-  ( ( B  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B ) )  ->  -.  ( B `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/) )
2221ex 435 . . . . . . . . . . . . 13  |-  ( B  e.  No  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  B )  ->  -.  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) ) )
2322adantl 467 . . . . . . . . . . . 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 74 . . . . . . . . . . 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 1202 . . . . . . . . . 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 430 . . . . . . . . 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 924 . . . . . . . 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 30317 . . . . . . . . . . . . 13  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
29 fnfvelrn 6034 . . . . . . . . . . . . . 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 2501 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 30316 . . . . . . . . . . . . . 14  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
3433sseld 3469 . . . . . . . . . . . . 13  |-  ( A  e.  No  ->  ( (/) 
e.  ran  A  ->  (/)  e.  { 1o ,  2o } ) )
3534adantr 466 . . . . . . . . . . . 12  |-  ( ( A  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A ) )  -> 
( (/)  e.  ran  A  -> 
(/)  e.  { 1o ,  2o } ) )
3632, 35syld 45 . . . . . . . . . . 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 181 . . . . . . . . . 10  |-  ( ( A  e.  No  /\  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A ) )  ->  -.  ( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  (/) )
38373ad2antl1 1167 . . . . . . . . 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 925 . . . . . . . 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 30128 . . . . . . . 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 665 . . . . . . 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 435 . . . . . 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 81 . . . . 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 220 . . . 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 218 . . 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 41 . 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 1175 . . . 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 215 . . 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 224 . 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 193 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 187    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   {crab 2786   (/)c0 3767   {cpr 4004   {ctp 4006   <.cop 4008   |^|cint 4258   class class class wbr 4426   ran crn 4855   Oncon0 5442    Fn wfn 5596   ` cfv 5601   1oc1o 7183   2oc2o 7184   Nocsur 30305   <scslt 30306   bdaycbday 30307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-1o 7190  df-2o 7191  df-no 30308  df-slt 30309  df-bday 30310
This theorem is referenced by:  nodense  30354
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