Theorem nodenselem5 30574

Description: Lemma for nodense 30578. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 30573 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 30559	. . . . . . . . 9 |- ( A e. No -> -. A <s A )
2		breq2 4406	. . . . . . . . . 10 |- ( A = B -> ( A <s A <-> A <s B ) )
3	2	notbid 296	. . . . . . . . 9 |- ( A = B -> ( -. A <s A <-> -. A <s B ) )
4	1, 3	syl5ibcom 224	. . . . . . . 8 |- ( A e. No -> ( A = B -> -. A <s B ) )
5	4	con2d 119	. . . . . . 7 |- ( A e. No -> ( A <s B -> -. A = B ) )
6	5	imp 431	. . . . . 6 |- ( ( A e. No /\ A <s B ) -> -. A = B )
7	6	ad2ant2rl 755	. . . . 5 |- ( ( ( A e. No /\ B e. No ) /\ ( dom A = dom B /\ A <s B ) ) -> -. A = B )
8		nofun 30536	. . . . . . . . 9 |- ( A e. No -> Fun A )
9		nofun 30536	. . . . . . . . 9 |- ( B e. No -> Fun B )
10		eqfunfv 5981	. . . . . . . . 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 480	. . . . . . . 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 296	. . . . . . 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 467	. . . . . 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 424	. . . . . . . . . . . 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 210	. . . . . . . . . . 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 432	. . . . . . . . . 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 2624	. . . . . . . . . . . . 13 |- ( ( A ` x ) =/= ( B ` x ) <-> -. ( A ` x ) = ( B ` x ) )
18	17	rexbii 2889	. . . . . . . . . . . 12 |- ( E. x e. dom A ( A ` x ) =/= ( B ` x ) <-> E. x e. dom A -. ( A ` x ) = ( B ` x ) )
19		rexnal 2836	. . . . . . . . . . . 12 |- ( E. x e. dom A -. ( A ` x ) = ( B ` x ) <-> -. A. x e. dom A ( A ` x ) = ( B ` x ) )
20	18, 19	bitri 253	. . . . . . . . . . 11 |- ( E. x e. dom A ( A ` x ) =/= ( B ` x ) <-> -. A. x e. dom A ( A ` x ) = ( B ` x ) )
21		nodenselem4 30573	. . . . . . . . . . . . . . 15 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
22		eloni 5433	. . . . . . . . . . . . . . 15 |- ( |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On -> Ord |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
23	21, 22	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> Ord |^| { a e. On | ( A ` a ) =/= ( B ` a ) } )
24	23	adantr 467	. . . . . . . . . . . . 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 30540	. . . . . . . . . . . . . 14 |- ( A e. No -> Ord dom A )
26	25	ad3antrrr 736	. . . . . . . . . . . . 13 |- ( ( ( ( A e. No /\ B e. No ) /\ A <s B ) /\ ( x e. dom A /\ ( A ` x ) =/= ( B ` x ) ) ) -> Ord dom A )
27		nodmon 30537	. . . . . . . . . . . . . . . . . . 19 |- ( A e. No -> dom A e. On )
28		onelon 5448	. . . . . . . . . . . . . . . . . . 19 |- ( ( dom A e. On /\ x e. dom A ) -> x e. On )
29	27, 28	sylan 474	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. No /\ x e. dom A ) -> x e. On )
30	29	ex 436	. . . . . . . . . . . . . . . . 17 |- ( A e. No -> ( x e. dom A -> x e. On ) )
31	30	ad2antrr 732	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> ( x e. dom A -> x e. On ) )
32	31	anim1d 568	. . . . . . . . . . . . . . 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 431	. . . . . . . . . . . . . 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 5865	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( A ` a ) = ( A ` x ) )
35		fveq2 5865	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( B ` a ) = ( B ` x ) )
36	34, 35	neeq12d 2685	. . . . . . . . . . . . . . 15 |- ( a = x -> ( ( A ` a ) =/= ( B ` a ) <-> ( A ` x ) =/= ( B ` x ) ) )
37	36	intminss 4261	. . . . . . . . . . . . . 14 |- ( ( x e. On /\ ( A ` x ) =/= ( B ` x ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } C_ x )
38	33, 37	syl 17	. . . . . . . . . . . . 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 764	. . . . . . . . . . . . 13 |- ( ( ( ( A e. No /\ B e. No ) /\ A <s B ) /\ ( x e. dom A /\ ( A ` x ) =/= ( B ` x ) ) ) -> x e. dom A )
40		ordtr2 5467	. . . . . . . . . . . . . 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 431	. . . . . . . . . . . . 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 1269	. . . . . . . . . . . 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 2880	. . . . . . . . . . 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 222	. . . . . . . . . 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 33	. . . . . . . . 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 612	. . . . . . . 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 81	. . . . . . 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 435	. . . . . 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 219	. . . . 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 436	. . 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 30535	. . . . 5 |- ( A e. No -> ( bday ` A ) = dom A )
53		bdayval 30535	. . . . 5 |- ( B e. No -> ( bday ` B ) = dom B )
54	52, 53	eqeqan12d 2467	. . . 4 |- ( ( A e. No /\ B e. No ) -> ( ( bday ` A ) = ( bday ` B ) <-> dom A = dom B ) )
55	54	anbi1d 711	. . 3 |- ( ( A e. No /\ B e. No ) -> ( ( ( bday ` A ) = ( bday ` B ) /\ A <s B ) <-> ( dom A = dom B /\ A <s B ) ) )
56	52	eleq2d 2514	. . . 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 467	. . 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 272	. 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 431	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 188   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   C_ wss 3404   |^|cint 4234   class class class wbr 4402   dom cdm 4834   Ord word 5422   Oncon0 5423   Fun wfun 5576   ` cfv 5582   Nocsur 30527   <scslt 30528   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-int 4235  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-slt 30531  df-bday 30532

This theorem is referenced by:  nodenselem6  30575  nodenselem8  30577  nodense  30578