Theorem nodenselem5 27826

Description: Lemma for nodense 27830. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 27825 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 27811	. . . . . . . . 9 |- ( A e. No -> -. A <s A )
2		breq2 4296	. . . . . . . . . 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 27790	. . . . . . . . 9 |- ( A e. No -> Fun A )
9		nofun 27790	. . . . . . . . 9 |- ( B e. No -> Fun B )
10		eqfunfv 5802	. . . . . . . . 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 2608	. . . . . . . . . . . . 13 |- ( ( A ` x ) =/= ( B ` x ) <-> -. ( A ` x ) = ( B ` x ) )
18	17	rexbii 2740	. . . . . . . . . . . 12 |- ( E. x e. dom A ( A ` x ) =/= ( B ` x ) <-> E. x e. dom A -. ( A ` x ) = ( B ` x ) )
19		rexnal 2726	. . . . . . . . . . . 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 27825	. . . . . . . . . . . . . . 15 |- ( ( ( A e. No /\ B e. No ) /\ A <s B ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On )
22		eloni 4729	. . . . . . . . . . . . . . 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 27794	. . . . . . . . . . . . . 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 27791	. . . . . . . . . . . . . . . . . . 19 |- ( A e. No -> dom A e. On )
28		onelon 4744	. . . . . . . . . . . . . . . . . . 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 5691	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( A ` a ) = ( A ` x ) )
35		fveq2 5691	. . . . . . . . . . . . . . . 16 |- ( a = x -> ( B ` a ) = ( B ` x ) )
36	34, 35	neeq12d 2623	. . . . . . . . . . . . . . 15 |- ( a = x -> ( ( A ` a ) =/= ( B ` a ) <-> ( A ` x ) =/= ( B ` x ) ) )
37	36	intminss 4154	. . . . . . . . . . . . . 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 4763	. . . . . . . . . . . . . 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 1219	. . . . . . . . . . . 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 2842	. . . . . . . . . . 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 27789	. . . . 5 |- ( A e. No -> ( bday ` A ) = dom A )
53		bdayval 27789	. . . . 5 |- ( B e. No -> ( bday ` B ) = dom B )
54	52, 53	eqeqan12d 2458	. . . 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 2510	. . . 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 1369   e. wcel 1756   =/= wne 2606   A.wral 2715   E.wrex 2716   {crab 2719   C_ wss 3328   |^|cint 4128   class class class wbr 4292   Ord word 4718   Oncon0 4719   dom cdm 4840   Fun wfun 5412   ` cfv 5418   Nocsur 27781   <scslt 27782   bdaycbday 27783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-1o 6920  df-2o 6921  df-no 27784  df-slt 27785  df-bday 27786

This theorem is referenced by:  nodenselem6  27827  nodenselem8  27829  nodense  27830