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Theorem nodenselem3 30572
Description: Lemma for nodense 30578. 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 30535 . . . 4  |-  ( B  e.  No  ->  ( bday `  B )  =  dom  B )
21adantl 468 . . 3  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( bday `  B
)  =  dom  B
)
32eleq2d 2514 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  ( bday `  B )  <->  ( bday `  A )  e.  dom  B ) )
4 bdayelon 30569 . . . 4  |-  ( bday `  A )  e.  On
5 nosgnn0 30545 . . . . . . . . 9  |-  -.  (/)  e.  { 1o ,  2o }
6 norn 30538 . . . . . . . . . . . 12  |-  ( B  e.  No  ->  ran  B 
C_  { 1o ,  2o } )
76adantr 467 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  ->  ran  B  C_  { 1o ,  2o } )
8 nofun 30536 . . . . . . . . . . . 12  |-  ( B  e.  No  ->  Fun  B )
9 fvelrn 6015 . . . . . . . . . . . 12  |-  ( ( Fun  B  /\  ( bday `  A )  e. 
dom  B )  -> 
( B `  ( bday `  A ) )  e.  ran  B )
108, 9sylan 474 . . . . . . . . . . 11  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  e.  ran  B )
117, 10sseldd 3433 . . . . . . . . . 10  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  e.  { 1o ,  2o } )
12 eleq1 2517 . . . . . . . . . 10  |-  ( ( B `  ( bday `  A ) )  =  (/)  ->  ( ( B `
 ( bday `  A
) )  e.  { 1o ,  2o }  <->  (/)  e.  { 1o ,  2o } ) )
1311, 12syl5ibcom 224 . . . . . . . . 9  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( ( B `  ( bday `  A )
)  =  (/)  ->  (/)  e.  { 1o ,  2o } ) )
145, 13mtoi 182 . . . . . . . 8  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  ->  -.  ( B `  ( bday `  A ) )  =  (/) )
1514neqned 2631 . . . . . . 7  |-  ( ( B  e.  No  /\  ( bday `  A )  e.  dom  B )  -> 
( B `  ( bday `  A ) )  =/=  (/) )
1615adantll 720 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( B `  ( bday `  A
) )  =/=  (/) )
17 fvnobday 30571 . . . . . . 7  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
1817ad2antrr 732 . . . . . 6  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( A `  ( bday `  A
) )  =  (/) )
1916, 18neeqtrrd 2698 . . . . 5  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( B `  ( bday `  A
) )  =/=  ( A `  ( bday `  A ) ) )
2019necomd 2679 . . . 4  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  ( A `  ( bday `  A
) )  =/=  ( B `  ( bday `  A ) ) )
21 fveq2 5865 . . . . . 6  |-  ( a  =  ( bday `  A
)  ->  ( A `  a )  =  ( A `  ( bday `  A ) ) )
22 fveq2 5865 . . . . . 6  |-  ( a  =  ( bday `  A
)  ->  ( B `  a )  =  ( B `  ( bday `  A ) ) )
2321, 22neeq12d 2685 . . . . 5  |-  ( a  =  ( bday `  A
)  ->  ( ( A `  a )  =/=  ( B `  a
)  <->  ( A `  ( bday `  A )
)  =/=  ( B `
 ( bday `  A
) ) ) )
2423rspcev 3150 . . . 4  |-  ( ( ( bday `  A
)  e.  On  /\  ( A `  ( bday `  A ) )  =/=  ( B `  ( bday `  A ) ) )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
254, 20, 24sylancr 669 . . 3  |-  ( ( ( A  e.  No  /\  B  e.  No )  /\  ( bday `  A
)  e.  dom  B
)  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a )
)
2625ex 436 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A
)  e.  dom  B  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
273, 26sylbid 219 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 371    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738    C_ wss 3404   (/)c0 3731   {cpr 3970   dom cdm 4834   ran crn 4835   Oncon0 5423   Fun wfun 5576   ` cfv 5582   1oc1o 7175   2oc2o 7176   Nocsur 30527   bdaycbday 30529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-1o 7182  df-2o 7183  df-no 30530  df-bday 30532
This theorem is referenced by:  nodenselem4  30573
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