Theorem nodenselem5 25553

Description: Lemma for nodense 25557. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 25552 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.

Step	Hyp	Ref	Expression
1		sltirr 25538	. . . . . . . . 9 |- ( A e. No -> -. A < s A )
2		breq2 4176	. . . . . . . . . 10 |- ( A = B -> ( A < s A <-> A < s B ) )
3	2	notbid 286	. . . . . . . . 9 |- ( A = B -> ( -. A < s A <-> -. A < s B ) )
4	1, 3	syl5ibcom 212	. . . . . . . 8 |- ( A e. No -> ( A = B -> -. A < s B ) )
5	4	con2d 109	. . . . . . 7 |- ( A e. No -> ( A < s B -> -. A = B ) )
6	5	imp 419	. . . . . 6 |- ( ( A e. No /\ A < s B ) -> -. A = B )
7	6	ad2ant2rl 730	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ ( dom A = dom B /\ A < s B ) ) -> -. A = B )
8		nofun 25517	. . . . . . . . 9 |- ( A e. No -> Fun A )
9		nofun 25517	. . . . . . . . 9 |- ( B e. No -> Fun B )
10		eqfunfv 5791	. . . . . . . . 9 |- ( ( Fun A /\ Fun B ) -> ( A = B <-> ( dom A = dom B /\ A. x e. dom A ( A ` x ) = ( B ` x ) ) ) )
11	8, 9, 10	syl2an 464	. . . . . . . 8 |- ( ( A e. No /\ B e. No ) -> ( A = B <-> ( dom A = dom B /\ A. x e. dom A ( A ` x ) = ( B ` x ) ) ) )
12	11	notbid 286	. . . . . . 7 |- ( ( A e. No /\ B e. No ) -> ( -. A = B <-> -. ( dom A = dom B /\ A. x e. dom A ( A ` x ) = ( B ` x ) ) ) )
13	12	adantr 452	. . . . . 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 412	. . . . . . . . . . . 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 ) ) )
15	14	biimpri 198	. . . . . . . . . . 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 ) ) )
16	15	impcom 420	. . . . . . . . . 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 2569	. . . . . . . . . . . . 13 |- ( ( A ` x ) =/= ( B ` x ) <-> -. ( A ` x ) = ( B ` x ) )
18	17	rexbii 2691	. . . . . . . . . . . 12 |- ( E. x e. dom A ( A ` x ) =/= ( B ` x ) <-> E. x e. dom A -. ( A ` x ) = ( B ` x ) )
19		rexnal 2677	. . . . . . . . . . . 12 |- ( E. x e. dom A -. ( A ` x ) = ( B ` x ) <-> -. A. x e. dom A ( A ` x ) = ( B ` x ) )
20	18, 19	bitri 241	. . . . . . . . . . 11 |- ( E. x e. dom A ( A ` x ) =/= ( B ` x ) <-> -. A. x e. dom A ( A ` x ) = ( B ` x ) )
21		nodenselem4 25552	. . . . . . . . . . . . . . 15 |- ( ( ( A e. No /\ B e. No ) /\ A < s B ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
22		eloni 4551	. . . . . . . . . . . . . . 15 |- ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On -> Ord |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
23	21, 22	syl 16	. . . . . . . . . . . . . 14 |- ( ( ( A e. No /\ B e. No ) /\ A < s B ) -> Ord |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
24	23	adantr 452	. . . . . . . . . . . . 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 25521	. . . . . . . . . . . . . 14 |- ( A e. No -> Ord dom A )
26	25	ad3antrrr 711	. . . . . . . . . . . . 13 |- ( ( ( ( A e. No /\ B e. No ) /\ A < s B ) /\ ( x e. dom A /\ ( A ` x ) =/= ( B ` x ) ) ) -> Ord dom A )
27		nodmon 25518	. . . . . . . . . . . . . . . . . . 19 |- ( A e. No -> dom A e. On )
28		onelon 4566	. . . . . . . . . . . . . . . . . . 19 |- ( ( dom A e. On /\ x e. dom A ) -> x e. On )
29	27, 28	sylan 458	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. No /\ x e. dom A ) -> x e. On )
30	29	ex 424	. . . . . . . . . . . . . . . . 17 |- ( A e. No -> ( x e. dom A -> x e. On ) )
31	30	ad2antrr 707	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. No /\ B e. No ) /\ A < s B ) -> ( x e. dom A -> x e. On ) )
32	31	anim1d 548	. . . . . . . . . . . . . . 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 ) ) ) )
33	32	imp 419	. . . . . . . . . . . . . 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 5687	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( A ` a ) = ( A ` x ) )
35		fveq2 5687	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( B ` a ) = ( B ` x ) )
36	34, 35	neeq12d 2582	. . . . . . . . . . . . . . 15 |- ( a = x -> ( ( A ` a ) =/= ( B ` a ) <-> ( A ` x ) =/= ( B ` x ) ) )
37	36	intminss 4036	. . . . . . . . . . . . . 14 |- ( ( x e. On /\ ( A ` x ) =/= ( B ` x ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } C_ x )
38	33, 37	syl 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 733	. . . . . . . . . . . . 13 |- ( ( ( ( A e. No /\ B e. No ) /\ A < s B ) /\ ( x e. dom A /\ ( A ` x ) =/= ( B ` x ) ) ) -> x e. dom A )
40		ordtr2 4585	. . . . . . . . . . . . . 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 ) )
41	40	imp 419	. . . . . . . . . . . . 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 )
42	24, 26, 38, 39, 41	syl22anc 1185	. . . . . . . . . . . 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 )
43	42	rexlimdvaa 2791	. . . . . . . . . . 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 ) )
44	20, 43	syl5bir 210	. . . . . . . . . 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 ) )
45	16, 44	syl5 30	. . . . . . . . 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 ) )
46	45	exp4b 591	. . . . . . . 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 ) ) ) )
47	46	com23 74	. . . . . . 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 ) ) ) )
48	47	imp32 423	. . . . . 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 ) )
49	13, 48	sylbid 207	. . . . 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 ) )
50	7, 49	mpd 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 )
51	50	ex 424	. . 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 25516	. . . . 5 |- ( A e. No -> ( bday ` A ) = dom A )
53		bdayval 25516	. . . . 5 |- ( B e. No -> ( bday ` B ) = dom B )
54	52, 53	eqeqan12d 2419	. . . 4 |- ( ( A e. No /\ B e. No ) -> ( ( bday ` A ) = ( bday ` B ) <-> dom A = dom B ) )
55	54	anbi1d 686	. . 3 |- ( ( A e. No /\ B e. No ) -> ( ( ( bday ` A ) = ( bday ` B ) /\ A < s B ) <-> ( dom A = dom B /\ A < s B ) ) )
56	52	eleq2d 2471	. . . 4 |- ( A e. No -> ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) <-> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. dom A ) )
57	56	adantr 452	. . 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 ) )
58	51, 55, 57	3imtr4d 260	. 2 |- ( ( A e. No /\ B e. No ) -> ( ( ( bday ` A ) = ( bday ` B ) /\ A < s B ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) ) )
59	58	imp 419	1 |- ( ( ( A e. No /\ B e. No ) /\ ( ( bday ` A ) = ( bday ` B ) /\ A < s B ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. ( bday ` A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   C_ wss 3280   |^|cint 4010   class class class wbr 4172   Ord word 4540   Oncon0 4541   dom cdm 4837   Fun wfun 5407   ` cfv 5413   Nocsur 25508   < scslt 25509   bdaycbday 25510

This theorem is referenced by:  nodenselem6  25554  nodenselem8  25556  nodense  25557

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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-2o 6684  df-no 25511  df-slt 25512  df-bday 25513