Theorem nodenselem5 27739

Description: Lemma for nodense 27743. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 27738 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 27724	. . . . . . . . 9 |- ( A e. No -> -. A <s A )
2		breq2 4293	. . . . . . . . . 10 |- ( A = B -> ( A <s A <-> A <s B ) )
3	2	notbid 294	. . . . . . . . 9 |- ( A = B -> ( -. A <s A <-> -. A <s B ) )
4	1, 3	syl5ibcom 220	. . . . . . . 8 |- ( A e. No -> ( A = B -> -. A <s B ) )
5	4	con2d 115	. . . . . . 7 |- ( A e. No -> ( A <s B -> -. A = B ) )
6	5	imp 429	. . . . . 6 |- ( ( A e. No /\ A <s B ) -> -. A = B )
7	6	ad2ant2rl 743	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ ( dom A = dom B /\ A <s B ) ) -> -. A = B )
8		nofun 27703	. . . . . . . . 9 |- ( A e. No -> Fun A )
9		nofun 27703	. . . . . . . . 9 |- ( B e. No -> Fun B )
10		eqfunfv 5799	. . . . . . . . 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 474	. . . . . . . 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 294	. . . . . . 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 462	. . . . . 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 422	. . . . . . . . . . . 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 206	. . . . . . . . . . 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 430	. . . . . . . . . 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 2606	. . . . . . . . . . . . 13 |- ( ( A ` x ) =/= ( B ` x ) <-> -. ( A ` x ) = ( B ` x ) )
18	17	rexbii 2738	. . . . . . . . . . . 12 |- ( E. x e. dom A ( A ` x ) =/= ( B ` x ) <-> E. x e. dom A -. ( A ` x ) = ( B ` x ) )
19		rexnal 2724	. . . . . . . . . . . 12 |- ( E. x e. dom A -. ( A ` x ) = ( B ` x ) <-> -. A. x e. dom A ( A ` x ) = ( B ` x ) )
20	18, 19	bitri 249	. . . . . . . . . . 11 |- ( E. x e. dom A ( A ` x ) =/= ( B ` x ) <-> -. A. x e. dom A ( A ` x ) = ( B ` x ) )
21		nodenselem4 27738	. . . . . . . . . . . . . . 15 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
22		eloni 4725	. . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . 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 27707	. . . . . . . . . . . . . 14 |- ( A e. No -> Ord dom A )
26	25	ad3antrrr 724	. . . . . . . . . . . . 13 |- ( ( ( ( A e. No /\ B e. No ) /\ A <s B ) /\ ( x e. dom A /\ ( A ` x ) =/= ( B ` x ) ) ) -> Ord dom A )
27		nodmon 27704	. . . . . . . . . . . . . . . . . . 19 |- ( A e. No -> dom A e. On )
28		onelon 4740	. . . . . . . . . . . . . . . . . . 19 |- ( ( dom A e. On /\ x e. dom A ) -> x e. On )
29	27, 28	sylan 468	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. No /\ x e. dom A ) -> x e. On )
30	29	ex 434	. . . . . . . . . . . . . . . . 17 |- ( A e. No -> ( x e. dom A -> x e. On ) )
31	30	ad2antrr 720	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> ( x e. dom A -> x e. On ) )
32	31	anim1d 561	. . . . . . . . . . . . . . 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 429	. . . . . . . . . . . . . 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 5688	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( A ` a ) = ( A ` x ) )
35		fveq2 5688	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( B ` a ) = ( B ` x ) )
36	34, 35	neeq12d 2621	. . . . . . . . . . . . . . 15 |- ( a = x -> ( ( A ` a ) =/= ( B ` a ) <-> ( A ` x ) =/= ( B ` x ) ) )
37	36	intminss 4151	. . . . . . . . . . . . . 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 750	. . . . . . . . . . . . 13 |- ( ( ( ( A e. No /\ B e. No ) /\ A <s B ) /\ ( x e. dom A /\ ( A ` x ) =/= ( B ` x ) ) ) -> x e. dom A )
40		ordtr2 4759	. . . . . . . . . . . . . 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 429	. . . . . . . . . . . . 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 1214	. . . . . . . . . . . 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 2840	. . . . . . . . . . 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 218	. . . . . . . . . 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 32	. . . . . . . . 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 604	. . . . . . . 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 78	. . . . . . 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 433	. . . . . 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 215	. . . . 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 434	. . 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 27702	. . . . 5 |- ( A e. No -> ( bday ` A ) = dom A )
53		bdayval 27702	. . . . 5 |- ( B e. No -> ( bday ` B ) = dom B )
54	52, 53	eqeqan12d 2456	. . . 4 |- ( ( A e. No /\ B e. No ) -> ( ( bday ` A ) = ( bday ` B ) <-> dom A = dom B ) )
55	54	anbi1d 699	. . 3 |- ( ( A e. No /\ B e. No ) -> ( ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) <-> ( dom A = dom B /\ A <s B ) ) )
56	52	eleq2d 2508	. . . 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 462	. . 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 268	. 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 429	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 184   /\ wa 369   = wceq 1364   e. wcel 1761   =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   C_ wss 3325   |^|cint 4125   class class class wbr 4289   Ord word 4714   Oncon0 4715   dom cdm 4836   Fun wfun 5409   ` cfv 5415   Nocsur 27694   <scslt 27695   bdaycbday 27696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-1o 6916  df-2o 6917  df-no 27697  df-slt 27698  df-bday 27699

This theorem is referenced by:  nodenselem6  27740  nodenselem8  27742  nodense  27743