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Theorem nodenselem3 25559
Description: Lemma for nodense 25565. 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 25524 . . . 4  |-  ( B  e.  No  ->  ( bday `  B )  =  dom  B )
21adantl 453 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( bday `  B
)  =  dom  B
)
32eleq2d 2479 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  <->  ( bday `  A )  e.  dom  B ) )
4 bdayelon 25556 . . . 4  |-  ( bday `  A )  e.  On
5 nosgnn0 25534 . . . . . . . . 9  |-  -.  (/)  e.  { 1o ,  2o }
6 norn 25527 . . . . . . . . . . . 12  |-  ( B  e.  No  ->  ran  B 
C_  { 1o ,  2o } )
76adantr 452 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  ->  ran  B  C_  { 1o ,  2o } )
8 nofun 25525 . . . . . . . . . . . 12  |-  ( B  e.  No  ->  Fun  B )
9 fvelrn 5833 . . . . . . . . . . . 12  |-  ( ( Fun  B  /\  ( bday `  A )  e. 
dom  B )  -> 
( B `  ( bday `  A ) )  e.  ran  B )
108, 9sylan 458 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  e.  ran  B )
117, 10sseldd 3317 . . . . . . . . . 10  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  e.  { 1o ,  2o } )
12 eleq1 2472 . . . . . . . . . 10  |-  ( ( B `  ( bday `  A ) )  =  (/)  ->  ( ( B `
 ( bday `  A
) )  e.  { 1o ,  2o }  <->  (/)  e.  { 1o ,  2o } ) )
1311, 12syl5ibcom 212 . . . . . . . . 9  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( ( B `  ( bday `  A )
)  =  (/)  ->  (/)  e.  { 1o ,  2o } ) )
145, 13mtoi 171 . . . . . . . 8  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  ->  -.  ( B `  ( bday `  A ) )  =  (/) )
1514neneqad 2645 . . . . . . 7  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  =/=  (/) )
1615adantll 695 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( B `  ( bday `  A
) )  =/=  (/) )
17 fvnobday 25558 . . . . . . 7  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
1817ad2antrr 707 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( A `  ( bday `  A
) )  =  (/) )
1916, 18neeqtrrd 2599 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( B `  ( bday `  A
) )  =/=  ( A `  ( bday `  A ) ) )
2019necomd 2658 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( A `  ( bday `  A
) )  =/=  ( B `  ( bday `  A ) ) )
21 fveq2 5695 . . . . . 6  |-  ( a  =  ( bday `  A
)  ->  ( A `  a )  =  ( A `  ( bday `  A ) ) )
22 fveq2 5695 . . . . . 6  |-  ( a  =  ( bday `  A
)  ->  ( B `  a )  =  ( B `  ( bday `  A ) ) )
2321, 22neeq12d 2590 . . . . 5  |-  ( a  =  ( bday `  A
)  ->  ( ( A `  a )  =/=  ( B `  a
)  <->  ( A `  ( bday `  A )
)  =/=  ( B `
 ( bday `  A
) ) ) )
2423rspcev 3020 . . . 4  |-  ( ( ( bday `  A
)  e.  On  /\  ( A `  ( bday `  A ) )  =/=  ( B `  ( bday `  A ) ) )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
254, 20, 24sylancr 645 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
2625ex 424 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  dom  B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
273, 26sylbid 207 1  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675    C_ wss 3288   (/)c0 3596   {cpr 3783   Oncon0 4549   dom cdm 4845   ran crn 4846   Fun wfun 5415   ` cfv 5421   1oc1o 6684   2oc2o 6685   Nocsur 25516   bdaycbday 25518
This theorem is referenced by:  nodenselem4  25560
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-1o 6691  df-2o 6692  df-no 25519  df-bday 25521
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