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Theorem nodenselem3 25363
Description: Lemma for nodense 25369. 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 25328 . . . 4  |-  ( B  e.  No  ->  ( bday `  B )  =  dom  B )
21adantl 453 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( bday `  B
)  =  dom  B
)
32eleq2d 2456 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  <->  ( bday `  A )  e.  dom  B ) )
4 bdayelon 25360 . . . 4  |-  ( bday `  A )  e.  On
5 nosgnn0 25338 . . . . . . . . 9  |-  -.  (/)  e.  { 1o ,  2o }
6 norn 25331 . . . . . . . . . . . 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 25329 . . . . . . . . . . . 12  |-  ( B  e.  No  ->  Fun  B )
9 fvelrn 5807 . . . . . . . . . . . 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 3294 . . . . . . . . . 10  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  e.  { 1o ,  2o } )
12 eleq1 2449 . . . . . . . . . 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 2622 . . . . . . 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 25362 . . . . . . 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 2576 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( B `  ( bday `  A
) )  =/=  ( A `  ( bday `  A ) ) )
2019necomd 2635 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( A `  ( bday `  A
) )  =/=  ( B `  ( bday `  A ) ) )
21 fveq2 5670 . . . . . 6  |-  ( a  =  ( bday `  A
)  ->  ( A `  a )  =  ( A `  ( bday `  A ) ) )
22 fveq2 5670 . . . . . 6  |-  ( a  =  ( bday `  A
)  ->  ( B `  a )  =  ( B `  ( bday `  A ) ) )
2321, 22neeq12d 2567 . . . . 5  |-  ( a  =  ( bday `  A
)  ->  ( ( A `  a )  =/=  ( B `  a
)  <->  ( A `  ( bday `  A )
)  =/=  ( B `
 ( bday `  A
) ) ) )
2423rspcev 2997 . . . 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 1717    =/= wne 2552   E.wrex 2652    C_ wss 3265   (/)c0 3573   {cpr 3760   Oncon0 4524   dom cdm 4820   ran crn 4821   Fun wfun 5390   ` cfv 5396   1oc1o 6655   2oc2o 6656   Nocsur 25320   bdaycbday 25322
This theorem is referenced by:  nodenselem4  25364
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-suc 4530  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-1o 6662  df-2o 6663  df-no 25323  df-bday 25325
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