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Theorem nodense 25557
Description: Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD) (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodense  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  E. x  e.  No  ( ( bday `  x )  e.  (
bday `  A )  /\  A < s x  /\  x < s B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem nodense
Dummy variables  a 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nodenselem6 25554 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e.  No )
2 bdayval 25516 . . . . 5  |-  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  No  ->  ( bday `  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  =  dom  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
31, 2syl 16 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( bday `  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  =  dom  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
4 dmres 5126 . . . . 5  |-  dom  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  =  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  i^i  dom 
A )
5 nodenselem5 25553 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )
)
6 bdayelon 25548 . . . . . . . . 9  |-  ( bday `  A )  e.  On
76onelssi 4649 . . . . . . . 8  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  ( bday `  A
)  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  ( bday `  A )
)
85, 7syl 16 . . . . . . 7  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  ( bday `  A )
)
9 bdayval 25516 . . . . . . . 8  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
109ad2antrr 707 . . . . . . 7  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( bday `  A )  =  dom  A )
118, 10sseqtrd 3344 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  dom  A )
12 df-ss 3294 . . . . . 6  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } 
C_  dom  A  <->  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  i^i  dom  A )  =  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
1311, 12sylib 189 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  i^i  dom  A )  =  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )
144, 13syl5eq 2448 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  dom  ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
153, 14eqtrd 2436 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( bday `  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  = 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
1615, 5eqeltrd 2478 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( bday `  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  e.  ( bday `  A
) )
17 nodenselem4 25552 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
1817adantrl 697 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
19 nodenselem8 25556 . . . . . . . . . . . . 13  |-  ( ( 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 ) ) )
2019biimpd 199 . . . . . . . . . . . 12  |-  ( ( 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 ) ) )
21203expia 1155 . . . . . . . . . . 11  |-  ( ( 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 ) ) ) )
2221imp32 423 . . . . . . . . . 10  |-  ( ( ( 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 ) )
2322simpld 446 . . . . . . . . 9  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o )
24 eqid 2404 . . . . . . . . 9  |-  (/)  =  (/)
2523, 24jctir 525 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (
( A `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } )  =  1o  /\  (/)  =  (/) ) )
26253mix1d 25123 . . . . . . 7  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (
( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o 
/\  (/)  =  (/) )  \/  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o 
/\  (/)  =  2o )  \/  ( ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  =  (/)  /\  (/)  =  2o ) ) )
27 fvex 5701 . . . . . . . 8  |-  ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
28 0ex 4299 . . . . . . . 8  |-  (/)  e.  _V
2927, 28brtp 25320 . . . . . . 7  |-  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }
(/) 
<->  ( ( ( A `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  =  1o  /\  (/)  =  (/) )  \/  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  1o  /\  (/)  =  2o )  \/  ( ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/)  /\  (/)  =  2o ) ) )
3026, 29sylibr 204 . . . . . 6  |-  ( ( ( 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 >. }
(/) )
3115fveq2d 5691 . . . . . . 7  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 ( bday `  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) ) )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) )
32 fvnobday 25550 . . . . . . . 8  |-  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  No  ->  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 ( bday `  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) ) )  =  (/) )
331, 32syl 16 . . . . . . 7  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 ( bday `  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) ) )  =  (/) )
3431, 33eqtr3d 2438 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  =  (/) )
3530, 34breqtrrd 4198 . . . . 5  |-  ( ( ( 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 >. }  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) )
36 fvres 5704 . . . . . . 7  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 y )  =  ( A `  y
) )
3736eqcomd 2409 . . . . . 6  |-  ( y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ->  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y ) )
3837rgen 2731 . . . . 5  |-  A. y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ( A `
 y )  =  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  y
)
3935, 38jctil 524 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( A. y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) )
40 raleq 2864 . . . . . 6  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A. y  e.  x  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  <->  A. y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ( A `
 y )  =  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  y
) ) )
41 fveq2 5687 . . . . . . 7  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  x )  =  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
42 fveq2 5687 . . . . . . 7  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 x )  =  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) )
4341, 42breq12d 4185 . . . . . 6  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  x )  <->  ( A `  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) ) )
4440, 43anbi12d 692 . . . . 5  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( A. y  e.  x  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  x
) )  <->  ( A. y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) ) )
4544rspcev 3012 . . . 4  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  ( A. y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  /\  ( A `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) } ) ) )  ->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  x
) ) )
4618, 39, 45syl2anc 643 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  x
) ) )
47 simpll 731 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  A  e.  No )
48 sltval 25515 . . . 4  |-  ( ( A  e.  No  /\  ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  No )  -> 
( A < s
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  <->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  x
) ) ) )
4947, 1, 48syl2anc 643 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( A < s ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  <->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  x
) ) ) )
5046, 49mpbird 224 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  A < s ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
5137adantl 453 . . . . . 6  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  (
( bday `  A )  =  ( bday `  B
)  /\  A < s B ) )  /\  y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  ( A `  y )  =  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 y ) )
52 nodenselem7 25555 . . . . . . 7  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (
y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A `  y )  =  ( B `  y ) ) )
5352imp 419 . . . . . 6  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  (
( bday `  A )  =  ( bday `  B
)  /\  A < s B ) )  /\  y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  ( A `  y )  =  ( B `  y ) )
5451, 53eqtr3d 2438 . . . . 5  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  (
( bday `  A )  =  ( bday `  B
)  /\  A < s B ) )  /\  y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  =  ( B `  y ) )
5554ralrimiva 2749 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  A. y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 y )  =  ( B `  y
) )
5622simprd 450 . . . . . . . 8  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )
5756, 24jctil 524 . . . . . . 7  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( (/)  =  (/)  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) )
58573mix3d 25125 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (
( (/)  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )  \/  ( (/)  =  1o  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )  \/  ( (/)  =  (/)  /\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o ) ) )
59 fvex 5701 . . . . . . 7  |-  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  e. 
_V
6028, 59brtp 25320 . . . . . 6  |-  ( (/) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  <-> 
( ( (/)  =  1o 
/\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  (/) )  \/  ( (/)  =  1o 
/\  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  =  2o )  \/  ( (/)  =  (/)  /\  ( B `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } )  =  2o ) ) )
6158, 60sylibr 204 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (/) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
6234, 61eqbrtrd 4192 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
63 raleq 2864 . . . . . 6  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( A. y  e.  x  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  y
)  =  ( B `
 y )  <->  A. y  e.  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 y )  =  ( B `  y
) ) )
64 fveq2 5687 . . . . . . 7  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( B `  x )  =  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )
6542, 64breq12d 4185 . . . . . 6  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x )  <->  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
6663, 65anbi12d 692 . . . . 5  |-  ( x  =  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  ->  ( ( A. y  e.  x  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  =  ( B `  y )  /\  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) )  <->  ( A. y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 y )  =  ( B `  y
)  /\  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) ) )
6766rspcev 3012 . . . 4  |-  ( (
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On  /\  ( A. y  e.  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `
 y )  =  ( B `  y
)  /\  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  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  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  y
)  =  ( B `
 y )  /\  ( ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) )
6818, 55, 62, 67syl12anc 1182 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  E. x  e.  On  ( A. y  e.  x  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  =  ( B `  y )  /\  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) )
69 simplr 732 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  B  e.  No )
70 sltval 25515 . . . 4  |-  ( ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  No  /\  B  e.  No )  ->  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) < s B  <->  E. x  e.  On  ( A. y  e.  x  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  =  ( B `  y )  /\  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) ) )
711, 69, 70syl2anc 643 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  (
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) < s B  <->  E. x  e.  On  ( A. y  e.  x  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  y )  =  ( B `  y )  /\  ( ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) ) )
7268, 71mpbird 224 . 2  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) <
s B )
73 fveq2 5687 . . . . 5  |-  ( x  =  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  ( bday `  x )  =  ( bday `  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) ) )
7473eleq1d 2470 . . . 4  |-  ( x  =  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  (
( bday `  x )  e.  ( bday `  A
)  <->  ( bday `  ( A  |`  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) } ) )  e.  ( bday `  A
) ) )
75 breq2 4176 . . . 4  |-  ( x  =  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  ( A < s x  <->  A < s ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) ) )
76 breq1 4175 . . . 4  |-  ( x  =  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  (
x < s B  <-> 
( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) < s B ) )
7774, 75, 763anbi123d 1254 . . 3  |-  ( x  =  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  ->  (
( ( bday `  x
)  e.  ( bday `  A )  /\  A < s x  /\  x < s B )  <->  ( ( bday `  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  e.  ( bday `  A
)  /\  A < s ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  /\  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) < s B ) ) )
7877rspcev 3012 . 2  |-  ( ( ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  e.  No  /\  (
( bday `  ( A  |` 
|^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) )  e.  ( bday `  A
)  /\  A < s ( A  |`  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )  /\  ( A  |`  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } ) < s B ) )  ->  E. x  e.  No  ( ( bday `  x
)  e.  ( bday `  A )  /\  A < s x  /\  x < s B ) )
791, 16, 50, 72, 78syl13anc 1186 1  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A < s B ) )  ->  E. x  e.  No  ( ( bday `  x )  e.  (
bday `  A )  /\  A < s x  /\  x < s B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    i^i cin 3279    C_ wss 3280   (/)c0 3588   {ctp 3776   <.cop 3777   |^|cint 4010   class class class wbr 4172   Oncon0 4541   dom cdm 4837    |` cres 4839   ` cfv 5413   1oc1o 6676   2oc2o 6677   Nocsur 25508   < scslt 25509   bdaycbday 25510
This theorem is referenced by:  nocvxminlem  25558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-2o 6684  df-no 25511  df-slt 25512  df-bday 25513
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