Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nodenselem5 Structured version   Unicode version

Theorem nodenselem5 27739
Description: Lemma for nodense 27743. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 27738 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 27724 . . . . . . . . 9  |-  ( A  e.  No  ->  -.  A <s A )
2 breq2 4293 . . . . . . . . . 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 743 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( dom  A  =  dom  B  /\  A <s B ) )  ->  -.  A  =  B )
8 nofun 27703 . . . . . . . . 9  |-  ( A  e.  No  ->  Fun  A )
9 nofun 27703 . . . . . . . . 9  |-  ( B  e.  No  ->  Fun  B )
10 eqfunfv 5799 . . . . . . . . 9  |-  ( ( Fun  A  /\  Fun  B )  ->  ( A  =  B  <->  ( dom  A  =  dom  B  /\  A. x  e.  dom  A ( A `  x )  =  ( B `  x ) ) ) )
118, 9, 10syl2an 474 . . . . . . . 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 462 . . . . . 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 2606 . . . . . . . . . . . . 13  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
1817rexbii 2738 . . . . . . . . . . . 12  |-  ( E. x  e.  dom  A
( A `  x
)  =/=  ( B `
 x )  <->  E. x  e.  dom  A  -.  ( A `  x )  =  ( B `  x ) )
19 rexnal 2724 . . . . . . . . . . . 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 27738 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  |^| { a  e.  On  |  ( A `  a )  =/=  ( B `  a ) }  e.  On )
22 eloni 4725 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . 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 27707 . . . . . . . . . . . . . 14  |-  ( A  e.  No  ->  Ord  dom 
A )
2625ad3antrrr 724 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  Ord  dom 
A )
27 nodmon 27704 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  No  ->  dom  A  e.  On )
28 onelon 4740 . . . . . . . . . . . . . . . . . . 19  |-  ( ( dom  A  e.  On  /\  x  e.  dom  A
)  ->  x  e.  On )
2927, 28sylan 468 . . . . . . . . . . . . . . . . . 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 720 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s
B )  ->  (
x  e.  dom  A  ->  x  e.  On ) )
3231anim1d 561 . . . . . . . . . . . . . . 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 5688 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( A `  a )  =  ( A `  x ) )
35 fveq2 5688 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( B `  a )  =  ( B `  x ) )
3634, 35neeq12d 2621 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( A `  a
)  =/=  ( B `
 a )  <->  ( A `  x )  =/=  ( B `  x )
) )
3736intminss 4151 . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  No  /\  B  e.  No )  /\  A <s B )  /\  ( x  e.  dom  A  /\  ( A `  x )  =/=  ( B `  x )
) )  ->  x  e.  dom  A )
40 ordtr2 4759 . . . . . . . . . . . . . 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 1214 . . . . . . . . . . . 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 2840 . . . . . . . . . . 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 604 . . . . . . . 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 27702 . . . . 5  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
53 bdayval 27702 . . . . 5  |-  ( B  e.  No  ->  ( bday `  B )  =  dom  B )
5452, 53eqeqan12d 2456 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  =  ( bday `  B )  <->  dom  A  =  dom  B ) )
5554anbi1d 699 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( ( bday `  A )  =  (
bday `  B )  /\  A <s B )  <->  ( dom  A  =  dom  B  /\  A <s B ) ) )
5652eleq2d 2508 . . . 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 462 . . 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 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717    C_ wss 3325   |^|cint 4125   class class class wbr 4289   Ord word 4714   Oncon0 4715   dom cdm 4836   Fun wfun 5409   ` cfv 5415   Nocsur 27694   <scslt 27695   bdaycbday 27696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-1o 6916  df-2o 6917  df-no 27697  df-slt 27698  df-bday 27699
This theorem is referenced by:  nodenselem6  27740  nodenselem8  27742  nodense  27743
  Copyright terms: Public domain W3C validator