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Theorem nodenselem5 29050
Description: Lemma for nodense 29054. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 29049 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )
)
Distinct variable groups:    A, a    B, a

Proof of Theorem nodenselem5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sltirr 29035 . . . . . . . . 9  |-  ( A  e.  No  ->  -.  A <s A )
2 breq2 4451 . . . . . . . . . 10  |-  ( A  =  B  ->  ( A <s A  <->  A <s B ) )
32notbid 294 . . . . . . . . 9  |-  ( A  =  B  ->  ( -.  A <s A  <->  -.  A <s B ) )
41, 3syl5ibcom 220 . . . . . . . 8  |-  ( A  e.  No  ->  ( A  =  B  ->  -.  A <s B ) )
54con2d 115 . . . . . . 7  |-  ( A  e.  No  ->  ( A <s B  ->  -.  A  =  B
) )
65imp 429 . . . . . 6  |-  ( ( A  e.  No  /\  A <s B )  ->  -.  A  =  B )
76ad2ant2rl 748 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  -.  A  =  B )
8 nofun 29014 . . . . . . . . 9  |-  ( A  e.  No  ->  Fun  A )
9 nofun 29014 . . . . . . . . 9  |-  ( B  e.  No  ->  Fun  B )
10 eqfunfv 5980 . . . . . . . . 9  |-  ( ( Fun  A  /\  Fun  B )  ->  ( A  =  B  <->  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x )  =  ( B `  x ) ) ) )
118, 9, 10syl2an 477 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <-> 
( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) ) ) )
1211notbid 294 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  <->  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A
( A `  x
)  =  ( B `
 x ) ) ) )
1312adantr 465 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  ( -.  A  =  B  <->  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A
( A `  x
)  =  ( B `
 x ) ) ) )
14 imnan 422 . . . . . . . . . . . 12  |-  ( ( dom  A  =  dom  B  ->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )  <->  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) ) )
1514biimpri 206 . . . . . . . . . . 11  |-  ( -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x )  =  ( B `  x ) )  -> 
( dom  A  =  dom  B  ->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) ) )
1615impcom 430 . . . . . . . . . 10  |-  ( ( dom  A  =  dom  B  /\  -.  ( dom 
A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) ) )  ->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
17 df-ne 2664 . . . . . . . . . . . . 13  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
1817rexbii 2965 . . . . . . . . . . . 12  |-  ( E. x  e.  dom  A
( A `  x
)  =/=  ( B `
 x )  <->  E. x  e.  dom  A  -.  ( A `  x )  =  ( B `  x ) )
19 rexnal 2912 . . . . . . . . . . . 12  |-  ( E. x  e.  dom  A  -.  ( A `  x
)  =  ( B `
 x )  <->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
2018, 19bitri 249 . . . . . . . . . . 11  |-  ( E. x  e.  dom  A
( A `  x
)  =/=  ( B `
 x )  <->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
21 nodenselem4 29049 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
22 eloni 4888 . . . . . . . . . . . . . . 15  |-  ( |^| { a  e.  On  | 
( A `  a
)  =/=  ( B `
 a ) }  e.  On  ->  Ord  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
2321, 22syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  Ord  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
2423adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  Ord  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) } )
25 nodmord 29018 . . . . . . . . . . . . . 14  |-  ( A  e.  No  ->  Ord  dom 
A )
2625ad3antrrr 729 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  Ord  dom 
A )
27 nodmon 29015 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  No  ->  dom  A  e.  On )
28 onelon 4903 . . . . . . . . . . . . . . . . . . 19  |-  ( ( dom  A  e.  On  /\  x  e.  dom  A
)  ->  x  e.  On )
2927, 28sylan 471 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  No  /\  x  e.  dom  A )  ->  x  e.  On )
3029ex 434 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  No  ->  (
x  e.  dom  A  ->  x  e.  On ) )
3130ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  (
x  e.  dom  A  ->  x  e.  On ) )
3231anim1d 564 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  (
( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
)  ->  ( x  e.  On  /\  ( A `
 x )  =/=  ( B `  x
) ) ) )
3332imp 429 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  (
x  e.  On  /\  ( A `  x )  =/=  ( B `  x ) ) )
34 fveq2 5866 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( A `  a )  =  ( A `  x ) )
35 fveq2 5866 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( B `  a )  =  ( B `  x ) )
3634, 35neeq12d 2746 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  x )  =/=  ( B `  x )
) )
3736intminss 4308 . . . . . . . . . . . . . 14  |-  ( ( x  e.  On  /\  ( A `  x )  =/=  ( B `  x ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
3833, 37syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  C_  x )
39 simprl 755 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  x  e.  dom  A )
40 ordtr2 4922 . . . . . . . . . . . . . 14  |-  ( ( Ord  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  /\  Ord  dom 
A )  ->  (
( |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  C_  x  /\  x  e.  dom  A )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
4140imp 429 . . . . . . . . . . . . 13  |-  ( ( ( Ord  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  /\  Ord  dom  A )  /\  ( |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  C_  x  /\  x  e.  dom  A ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A )
4224, 26, 38, 39, 41syl22anc 1229 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A )
4342rexlimdvaa 2956 . . . . . . . . . . 11  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  ( E. x  e.  dom  A ( A `  x
)  =/=  ( B `
 x )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
4420, 43syl5bir 218 . . . . . . . . . 10  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  ( -.  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
4516, 44syl5 32 . . . . . . . . 9  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  (
( dom  A  =  dom  B  /\  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
4645exp4b 607 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A <s
B  ->  ( dom  A  =  dom  B  -> 
( -.  ( dom 
A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) )  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  dom  A ) ) ) )
4746com23 78 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( dom  A  =  dom  B  ->  ( A <s B  -> 
( -.  ( dom 
A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) )  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  dom  A ) ) ) )
4847imp32 433 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  ( -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x
)  =  ( B `
 x ) )  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  dom  A ) )
4913, 48sylbid 215 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  ( -.  A  =  B  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
507, 49mpd 15 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `
 a )  =/=  ( B `  a
) }  e.  dom  A )
5150ex 434 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( dom  A  =  dom  B  /\  A <s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
52 bdayval 29013 . . . . 5  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
53 bdayval 29013 . . . . 5  |-  ( B  e.  No  ->  ( bday `  B )  =  dom  B )
5452, 53eqeqan12d 2490 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  =  ( bday `  B )  <->  dom  A  =  dom  B ) )
5554anbi1d 704 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( ( bday `  A )  =  (
bday `  B )  /\  A <s B )  <->  ( dom  A  =  dom  B  /\  A <s B ) ) )
5652eleq2d 2537 . . . 4  |-  ( A  e.  No  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )  <->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
5756adantr 465 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )  <->  |^|
{ a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  dom  A ) )
5851, 55, 573imtr4d 268 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( ( bday `  A )  =  (
bday `  B )  /\  A <s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )
) )
5958imp 429 1  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
bday `  B )  /\  A <s B ) )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  ( bday `  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    C_ wss 3476   |^|cint 4282   class class class wbr 4447   Ord word 4877   Oncon0 4878   dom cdm 4999   Fun wfun 5582   ` cfv 5588   Nocsur 29005   <scslt 29006   bdaycbday 29007
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 6576
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-1o 7130  df-2o 7131  df-no 29008  df-slt 29009  df-bday 29010
This theorem is referenced by:  nodenselem6  29051  nodenselem8  29053  nodense  29054
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