Theorem nodenselem5 29050

Description: Lemma for nodense 29054. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 29049 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 29035	. . . . . . . . 9 |- ( A e. No -> -. A <s A )
2		breq2 4451	. . . . . . . . . 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 748	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ ( dom A = dom B /\ A <s B ) ) -> -. A = B )
8		nofun 29014	. . . . . . . . 9 |- ( A e. No -> Fun A )
9		nofun 29014	. . . . . . . . 9 |- ( B e. No -> Fun B )
10		eqfunfv 5980	. . . . . . . . 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 477	. . . . . . . 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 465	. . . . . 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 2664	. . . . . . . . . . . . 13 |- ( ( A ` x ) =/= ( B ` x ) <-> -. ( A ` x ) = ( B ` x ) )
18	17	rexbii 2965	. . . . . . . . . . . 12 |- ( E. x e. dom A ( A ` x ) =/= ( B ` x ) <-> E. x e. dom A -. ( A ` x ) = ( B ` x ) )
19		rexnal 2912	. . . . . . . . . . . 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 29049	. . . . . . . . . . . . . . 15 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
22		eloni 4888	. . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . 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 29018	. . . . . . . . . . . . . 14 |- ( A e. No -> Ord dom A )
26	25	ad3antrrr 729	. . . . . . . . . . . . 13 |- ( ( ( ( A e. No /\ B e. No ) /\ A <s B ) /\ ( x e. dom A /\ ( A ` x ) =/= ( B ` x ) ) ) -> Ord dom A )
27		nodmon 29015	. . . . . . . . . . . . . . . . . . 19 |- ( A e. No -> dom A e. On )
28		onelon 4903	. . . . . . . . . . . . . . . . . . 19 |- ( ( dom A e. On /\ x e. dom A ) -> x e. On )
29	27, 28	sylan 471	. . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> ( x e. dom A -> x e. On ) )
32	31	anim1d 564	. . . . . . . . . . . . . . 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 5866	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( A ` a ) = ( A ` x ) )
35		fveq2 5866	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( B ` a ) = ( B ` x ) )
36	34, 35	neeq12d 2746	. . . . . . . . . . . . . . 15 |- ( a = x -> ( ( A ` a ) =/= ( B ` a ) <-> ( A ` x ) =/= ( B ` x ) ) )
37	36	intminss 4308	. . . . . . . . . . . . . 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 755	. . . . . . . . . . . . 13 |- ( ( ( ( A e. No /\ B e. No ) /\ A <s B ) /\ ( x e. dom A /\ ( A ` x ) =/= ( B ` x ) ) ) -> x e. dom A )
40		ordtr2 4922	. . . . . . . . . . . . . 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 1229	. . . . . . . . . . . 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 2956	. . . . . . . . . . 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 607	. . . . . . . 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 29013	. . . . 5 |- ( A e. No -> ( bday ` A ) = dom A )
53		bdayval 29013	. . . . 5 |- ( B e. No -> ( bday ` B ) = dom B )
54	52, 53	eqeqan12d 2490	. . . 4 |- ( ( A e. No /\ B e. No ) -> ( ( bday ` A ) = ( bday ` B ) <-> dom A = dom B ) )
55	54	anbi1d 704	. . 3 |- ( ( A e. No /\ B e. No ) -> ( ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) <-> ( dom A = dom B /\ A <s B ) ) )
56	52	eleq2d 2537	. . . 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 465	. . 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 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   C_ wss 3476   |^|cint 4282   class class class wbr 4447   Ord word 4877   Oncon0 4878   dom cdm 4999   Fun wfun 5582   ` cfv 5588   Nocsur 29005   <scslt 29006   bdaycbday 29007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-1o 7130  df-2o 7131  df-no 29008  df-slt 29009  df-bday 29010

This theorem is referenced by:  nodenselem6  29051  nodenselem8  29053  nodense  29054