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Theorem nodenselem3 29415
Description: Lemma for nodense 29421. If one surreal is older than another, then there is an ordinal at which they are not equal. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
Distinct variable groups:    A, a    B, a

Proof of Theorem nodenselem3
StepHypRef Expression
1 bdayval 29380 . . . 4  |-  ( B  e.  No  ->  ( bday `  B )  =  dom  B )
21adantl 466 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( bday `  B
)  =  dom  B
)
32eleq2d 2511 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  <->  ( bday `  A )  e.  dom  B ) )
4 bdayelon 29412 . . . 4  |-  ( bday `  A )  e.  On
5 nosgnn0 29390 . . . . . . . . 9  |-  -.  (/)  e.  { 1o ,  2o }
6 norn 29383 . . . . . . . . . . . 12  |-  ( B  e.  No  ->  ran  B 
C_  { 1o ,  2o } )
76adantr 465 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  ->  ran  B  C_  { 1o ,  2o } )
8 nofun 29381 . . . . . . . . . . . 12  |-  ( B  e.  No  ->  Fun  B )
9 fvelrn 6006 . . . . . . . . . . . 12  |-  ( ( Fun  B  /\  ( bday `  A )  e. 
dom  B )  -> 
( B `  ( bday `  A ) )  e.  ran  B )
108, 9sylan 471 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  e.  ran  B )
117, 10sseldd 3488 . . . . . . . . . 10  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  e.  { 1o ,  2o } )
12 eleq1 2513 . . . . . . . . . 10  |-  ( ( B `  ( bday `  A ) )  =  (/)  ->  ( ( B `
 ( bday `  A
) )  e.  { 1o ,  2o }  <->  (/)  e.  { 1o ,  2o } ) )
1311, 12syl5ibcom 220 . . . . . . . . 9  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( ( B `  ( bday `  A )
)  =  (/)  ->  (/)  e.  { 1o ,  2o } ) )
145, 13mtoi 178 . . . . . . . 8  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  ->  -.  ( B `  ( bday `  A ) )  =  (/) )
1514neqned 2644 . . . . . . 7  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  =/=  (/) )
1615adantll 713 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( B `  ( bday `  A
) )  =/=  (/) )
17 fvnobday 29414 . . . . . . 7  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
1817ad2antrr 725 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( A `  ( bday `  A
) )  =  (/) )
1916, 18neeqtrrd 2741 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( B `  ( bday `  A
) )  =/=  ( A `  ( bday `  A ) ) )
2019necomd 2712 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( A `  ( bday `  A
) )  =/=  ( B `  ( bday `  A ) ) )
21 fveq2 5853 . . . . . 6  |-  ( a  =  ( bday `  A
)  ->  ( A `  a )  =  ( A `  ( bday `  A ) ) )
22 fveq2 5853 . . . . . 6  |-  ( a  =  ( bday `  A
)  ->  ( B `  a )  =  ( B `  ( bday `  A ) ) )
2321, 22neeq12d 2720 . . . . 5  |-  ( a  =  ( bday `  A
)  ->  ( ( A `  a )  =/=  ( B `  a
)  <->  ( A `  ( bday `  A )
)  =/=  ( B `
 ( bday `  A
) ) ) )
2423rspcev 3194 . . . 4  |-  ( ( ( bday `  A
)  e.  On  /\  ( A `  ( bday `  A ) )  =/=  ( B `  ( bday `  A ) ) )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
254, 20, 24sylancr 663 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
2625ex 434 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  dom  B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
273, 26sylbid 215 1  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   E.wrex 2792    C_ wss 3459   (/)c0 3768   {cpr 4013   Oncon0 4865   dom cdm 4986   ran crn 4987   Fun wfun 5569   ` cfv 5575   1oc1o 7122   2oc2o 7123   Nocsur 29372   bdaycbday 29374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-1o 7129  df-2o 7130  df-no 29375  df-bday 29377
This theorem is referenced by:  nodenselem4  29416
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