Theorem onfrALTlem2 38761

Description: Lemma for onfrALT 38764. (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 477	. . . . . . . . . . . 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	2a1i 12	. . . . . . . . . . 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 3834	. . . . . . . . . . . 12 |- ( a i^i y ) C_ y
4	3	sseli 3599	. . . . . . . . . . 11 |- ( z e. ( a i^i y ) -> z e. y )
5	2, 4	syl8 76	. . . . . . . . . 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 3833	. . . . . . . . . . . . 13 |- ( a i^i y ) C_ a
7	6	sseli 3599	. . . . . . . . . . . 12 |- ( z e. ( a i^i y ) -> z e. a )
8	2, 7	syl8 76	. . . . . . . . . . 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 473	. . . . . . . . . . . . . . 15 |- ( ( a C_ On /\ a =/= (/) ) -> a C_ On )
10		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. a )
11		ssel 3597	. . . . . . . . . . . . . . 15 |- ( a C_ On -> ( x e. a -> x e. On ) )
12	9, 10, 11	syl2im 40	. . . . . . . . . . . . . 14 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. On ) )
13		eloni 5733	. . . . . . . . . . . . . 14 |- ( x e. On -> Ord x )
14	12, 13	syl6 35	. . . . . . . . . . . . 13 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> Ord x ) )
15		ordtr 5737	. . . . . . . . . . . . 13 |- ( Ord x -> Tr x )
16	14, 15	syl6 35	. . . . . . . . . . . 12 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> Tr x ) )
17		simpll 790	. . . . . . . . . . . . . 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	2a1i 12	. . . . . . . . . . . . 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 3834	. . . . . . . . . . . . . 14 |- ( a i^i x ) C_ x
20	19	sseli 3599	. . . . . . . . . . . . 13 |- ( y e. ( a i^i x ) -> y e. x )
21	18, 20	syl8 76	. . . . . . . . . . . 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 4759	. . . . . . . . . . . . 13 |- ( Tr x -> ( ( z e. y /\ y e. x ) -> z e. x ) )
23	22	expcomd 454	. . . . . . . . . . . 12 |- ( Tr x -> ( y e. x -> ( z e. y -> z e. x ) ) )
24	16, 21, 5, 23	ee233 38725	. . . . . . . . . . 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 3796	. . . . . . . . . . . 12 |- ( z e. ( a i^i x ) <-> ( z e. a /\ z e. x ) )
26	25	simplbi2 655	. . . . . . . . . . 11 |- ( z e. a -> ( z e. x -> z e. ( a i^i x ) ) )
27	8, 24, 26	ee33 38727	. . . . . . . . . 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 3796	. . . . . . . . . . 11 |- ( z e. ( ( a i^i x ) i^i y ) <-> ( z e. ( a i^i x ) /\ z e. y ) )
29	28	simplbi2com 657	. . . . . . . . . 10 |- ( z e. y -> ( z e. ( a i^i x ) -> z e. ( ( a i^i x ) i^i y ) ) )
30	5, 27, 29	ee33 38727	. . . . . . . . 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 633	. . . . . . . 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 38760	. . . . . . 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 3591	. . . . . . . 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 218	. . . . . . 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 76	. . . . . 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 477	. . . . . . 7 |- ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( ( a i^i x ) i^i y ) = (/) )
37	36	2a1i 12	. . . . . 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 3975	. . . . . . 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 450	. . . . . 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 38727	. . . . 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 473	. . . . . . 7 |- ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> y e. ( a i^i x ) )
42	41	2a1i 12	. . . . . 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 3833	. . . . . . 7 |- ( a i^i x ) C_ a
44	43	sseli 3599	. . . . . 6 |- ( y e. ( a i^i x ) -> y e. a )
45	42, 44	syl8 76	. . . . 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		pm3.21 464	. . . . 5 |- ( ( a i^i y ) = (/) -> ( y e. a -> ( y e. a /\ ( a i^i y ) = (/) ) ) )
47	40, 45, 46	ee33 38727	. . . 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 1857	. . 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 38759	. . . 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 2918	. . . 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 241	. . 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 1761	. . 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 70	. 2 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> E. y ( y e. a /\ ( a i^i y ) = (/) ) ) )
54		df-rex 2918	. 2 |- ( E. y e. a ( a i^i y ) = (/) <-> E. y ( y e. a /\ ( a i^i y ) = (/) ) )
55	53, 54	syl6ibr 242	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 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   Tr wtr 4752   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  onfrALT  38764