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Theorem nodenselem5 25553
Description: Lemma for nodense 25557. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 25552 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 25538 . . . . . . . . 9  |-  ( A  e.  No  ->  -.  A < s A )
2 breq2 4176 . . . . . . . . . 10  |-  ( A  =  B  ->  ( A < s A  <->  A < s B ) )
32notbid 286 . . . . . . . . 9  |-  ( A  =  B  ->  ( -.  A < s A  <->  -.  A < s B ) )
41, 3syl5ibcom 212 . . . . . . . 8  |-  ( A  e.  No  ->  ( A  =  B  ->  -.  A < s B ) )
54con2d 109 . . . . . . 7  |-  ( A  e.  No  ->  ( A < s B  ->  -.  A  =  B
) )
65imp 419 . . . . . 6  |-  ( ( A  e.  No  /\  A < s B )  ->  -.  A  =  B )
76ad2ant2rl 730 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A < s B ) )  ->  -.  A  =  B )
8 nofun 25517 . . . . . . . . 9  |-  ( A  e.  No  ->  Fun  A )
9 nofun 25517 . . . . . . . . 9  |-  ( B  e.  No  ->  Fun  B )
10 eqfunfv 5791 . . . . . . . . 9  |-  ( ( Fun  A  /\  Fun  B )  ->  ( A  =  B  <->  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x )  =  ( B `  x ) ) ) )
118, 9, 10syl2an 464 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <-> 
( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) ) ) )
1211notbid 286 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( -.  A  =  B  <->  -.  ( dom  A  =  dom  B  /\  A. x  e.  dom  A
( A `  x
)  =  ( B `
 x ) ) ) )
1312adantr 452 . . . . . 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 412 . . . . . . . . . . . 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 198 . . . . . . . . . . 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 420 . . . . . . . . . 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 2569 . . . . . . . . . . . . 13  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
1817rexbii 2691 . . . . . . . . . . . 12  |-  ( E. x  e.  dom  A
( A `  x
)  =/=  ( B `
 x )  <->  E. x  e.  dom  A  -.  ( A `  x )  =  ( B `  x ) )
19 rexnal 2677 . . . . . . . . . . . 12  |-  ( E. x  e.  dom  A  -.  ( A `  x
)  =  ( B `
 x )  <->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
2018, 19bitri 241 . . . . . . . . . . 11  |-  ( E. x  e.  dom  A
( A `  x
)  =/=  ( B `
 x )  <->  -.  A. x  e.  dom  A ( A `
 x )  =  ( B `  x
) )
21 nodenselem4 25552 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
22 eloni 4551 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 25521 . . . . . . . . . . . . . 14  |-  ( A  e.  No  ->  Ord  dom 
A )
2625ad3antrrr 711 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  Ord  dom 
A )
27 nodmon 25518 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  No  ->  dom  A  e.  On )
28 onelon 4566 . . . . . . . . . . . . . . . . . . 19  |-  ( ( dom  A  e.  On  /\  x  e.  dom  A
)  ->  x  e.  On )
2927, 28sylan 458 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  No  /\  x  e.  dom  A )  ->  x  e.  On )
3029ex 424 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  No  ->  (
x  e.  dom  A  ->  x  e.  On ) )
3130ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  (
x  e.  dom  A  ->  x  e.  On ) )
3231anim1d 548 . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( A `  a )  =  ( A `  x ) )
35 fveq2 5687 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( B `  a )  =  ( B `  x ) )
3634, 35neeq12d 2582 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  x )  =/=  ( B `  x )
) )
3736intminss 4036 . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  x  e.  dom  A )
40 ordtr2 4585 . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 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 2791 . . . . . . . . . . 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 210 . . . . . . . . . 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 30 . . . . . . . . 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 591 . . . . . . . 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 74 . . . . . . 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 423 . . . . . 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 207 . . . . 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 424 . . 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 25516 . . . . 5  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
53 bdayval 25516 . . . . 5  |-  ( B  e.  No  ->  ( bday `  B )  =  dom  B )
5452, 53eqeqan12d 2419 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  =  ( bday `  B )  <->  dom  A  =  dom  B ) )
5554anbi1d 686 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( ( bday `  A )  =  (
bday `  B )  /\  A < s B )  <->  ( dom  A  =  dom  B  /\  A < s B ) ) )
5652eleq2d 2471 . . . 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 452 . . 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 260 . 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 419 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   |^|cint 4010   class class class wbr 4172   Ord word 4540   Oncon0 4541   dom cdm 4837   Fun wfun 5407   ` cfv 5413   Nocsur 25508   < scslt 25509   bdaycbday 25510
This theorem is referenced by:  nodenselem6  25554  nodenselem8  25556  nodense  25557
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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