Theorem onfrALTlem2 31141

Description: Lemma for onfrALT 31144. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALTlem2	|- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> E. y e. a ( a i^i y ) = (/) ) )

Distinct variable groups:   y, a   x, y

Proof of Theorem onfrALTlem2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 461	. . . . . . . . . . . 12 |- ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> z e. ( a i^i y ) )
2	1	a1ii 27	. . . . . . . . . . 11 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> z e. ( a i^i y ) ) ) )
3		inss2 3566	. . . . . . . . . . . 12 |- ( a i^i y ) C_ y
4	3	sseli 3347	. . . . . . . . . . 11 |- ( z e. ( a i^i y ) -> z e. y )
5	2, 4	syl8 70	. . . . . . . . . 10 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> z e. y ) ) )
6		inss1 3565	. . . . . . . . . . . . 13 |- ( a i^i y ) C_ a
7	6	sseli 3347	. . . . . . . . . . . 12 |- ( z e. ( a i^i y ) -> z e. a )
8	2, 7	syl8 70	. . . . . . . . . . 11 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> z e. a ) ) )
9		simpl 457	. . . . . . . . . . . . . . 15 |- ( ( a C_ On /\ a =/= (/) ) -> a C_ On )
10		simpl 457	. . . . . . . . . . . . . . 15 |- ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. a )
11		ssel 3345	. . . . . . . . . . . . . . 15 |- ( a C_ On -> ( x e. a -> x e. On ) )
12	9, 10, 11	syl2im 38	. . . . . . . . . . . . . 14 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. On ) )
13		eloni 4724	. . . . . . . . . . . . . 14 |- ( x e. On -> Ord x )
14	12, 13	syl6 33	. . . . . . . . . . . . 13 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> Ord x ) )
15		ordtr 4728	. . . . . . . . . . . . 13 |- ( Ord x -> Tr x )
16	14, 15	syl6 33	. . . . . . . . . . . 12 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> Tr x ) )
17		simpll 753	. . . . . . . . . . . . . 14 |- ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> y e. ( a i^i x ) )
18	17	a1ii 27	. . . . . . . . . . . . 13 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> y e. ( a i^i x ) ) ) )
19		inss2 3566	. . . . . . . . . . . . . 14 |- ( a i^i x ) C_ x
20	19	sseli 3347	. . . . . . . . . . . . 13 |- ( y e. ( a i^i x ) -> y e. x )
21	18, 20	syl8 70	. . . . . . . . . . . 12 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> y e. x ) ) )
22		trel 4387	. . . . . . . . . . . . 13 |- ( Tr x -> ( ( z e. y /\ y e. x ) -> z e. x ) )
23	22	expcomd 438	. . . . . . . . . . . 12 |- ( Tr x -> ( y e. x -> ( z e. y -> z e. x ) ) )
24	16, 21, 5, 23	ee233 31111	. . . . . . . . . . 11 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> z e. x ) ) )
25		elin 3534	. . . . . . . . . . . 12 |- ( z e. ( a i^i x ) <-> ( z e. a /\ z e. x ) )
26	25	simplbi2 625	. . . . . . . . . . 11 |- ( z e. a -> ( z e. x -> z e. ( a i^i x ) ) )
27	8, 24, 26	ee33 31113	. . . . . . . . . 10 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> z e. ( a i^i x ) ) ) )
28		elin 3534	. . . . . . . . . . 11 |- ( z e. ( ( a i^i x ) i^i y ) <-> ( z e. ( a i^i x ) /\ z e. y ) )
29	28	simplbi2com 627	. . . . . . . . . 10 |- ( z e. y -> ( z e. ( a i^i x ) -> z e. ( ( a i^i x ) i^i y ) ) )
30	5, 27, 29	ee33 31113	. . . . . . . . 9 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) -> z e. ( ( a i^i x ) i^i y ) ) ) )
31	30	exp4a 606	. . . . . . . 8 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( z e. ( a i^i y ) -> z e. ( ( a i^i x ) i^i y ) ) ) ) )
32	31	ggen31 31140	. . . . . . 7 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> A. z ( z e. ( a i^i y ) -> z e. ( ( a i^i x ) i^i y ) ) ) ) )
33		dfss2 3340	. . . . . . . 8 |- ( ( a i^i y ) C_ ( ( a i^i x ) i^i y ) <-> A. z ( z e. ( a i^i y ) -> z e. ( ( a i^i x ) i^i y ) ) )
34	33	biimpri 206	. . . . . . 7 |- ( A. z ( z e. ( a i^i y ) -> z e. ( ( a i^i x ) i^i y ) ) -> ( a i^i y ) C_ ( ( a i^i x ) i^i y ) )
35	32, 34	syl8 70	. . . . . 6 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( a i^i y ) C_ ( ( a i^i x ) i^i y ) ) ) )
36		simpr 461	. . . . . . 7 |- ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( ( a i^i x ) i^i y ) = (/) )
37	36	a1ii 27	. . . . . 6 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( ( a i^i x ) i^i y ) = (/) ) ) )
38		sseq0 3664	. . . . . . 7 |- ( ( ( a i^i y ) C_ ( ( a i^i x ) i^i y ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( a i^i y ) = (/) )
39	38	ex 434	. . . . . 6 |- ( ( a i^i y ) C_ ( ( a i^i x ) i^i y ) -> ( ( ( a i^i x ) i^i y ) = (/) -> ( a i^i y ) = (/) ) )
40	35, 37, 39	ee33 31113	. . . . 5 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( a i^i y ) = (/) ) ) )
41		simpl 457	. . . . . . 7 |- ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> y e. ( a i^i x ) )
42	41	a1ii 27	. . . . . 6 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> y e. ( a i^i x ) ) ) )
43		inss1 3565	. . . . . . 7 |- ( a i^i x ) C_ a
44	43	sseli 3347	. . . . . 6 |- ( y e. ( a i^i x ) -> y e. a )
45	42, 44	syl8 70	. . . . 5 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> y e. a ) ) )
46		pm3.21 448	. . . . 5 |- ( ( a i^i y ) = (/) -> ( y e. a -> ( y e. a /\ ( a i^i y ) = (/) ) ) )
47	40, 45, 46	ee33 31113	. . . 4 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( y e. a /\ ( a i^i y ) = (/) ) ) ) )
48	47	alrimdv 1687	. . 3 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> A. y ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( y e. a /\ ( a i^i y ) = (/) ) ) ) )
49		onfrALTlem3 31139	. . . 4 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )
50		df-rex 2716	. . . 4 |- ( E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
51	49, 50	syl6ib 226	. . 3 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) ) )
52		exim 1623	. . 3 |- ( A. y ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( y e. a /\ ( a i^i y ) = (/) ) ) -> ( E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> E. y ( y e. a /\ ( a i^i y ) = (/) ) ) )
53	48, 51, 52	syl6c 64	. 2 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> E. y ( y e. a /\ ( a i^i y ) = (/) ) ) )
54		df-rex 2716	. 2 |- ( E. y e. a ( a i^i y ) = (/) <-> E. y ( y e. a /\ ( a i^i y ) = (/) ) )
55	53, 54	syl6ibr 227	1 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> E. y e. a ( a i^i y ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   A.wal 1367   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2601   E.wrex 2711   i^i cin 3322   C_ wss 3323   (/)c0 3632   Tr wtr 4380   Ord word 4713   Oncon0 4714

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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-tr 4381  df-eprel 4627  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718

This theorem is referenced by:  onfrALT  31144