Theorem onfrALTlem2 36906

Description: Lemma for onfrALT 36909. (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 463	. . . . . . . . . . . 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 3652	. . . . . . . . . . . 12 |- ( a i^i y ) C_ y
4	3	sseli 3427	. . . . . . . . . . 11 |- ( z e. ( a i^i y ) -> z e. y )
5	2, 4	syl8 72	. . . . . . . . . 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 3651	. . . . . . . . . . . . 13 |- ( a i^i y ) C_ a
7	6	sseli 3427	. . . . . . . . . . . 12 |- ( z e. ( a i^i y ) -> z e. a )
8	2, 7	syl8 72	. . . . . . . . . . 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 459	. . . . . . . . . . . . . . 15 |- ( ( a C_ On /\ a =/= (/) ) -> a C_ On )
10		simpl 459	. . . . . . . . . . . . . . 15 |- ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. a )
11		ssel 3425	. . . . . . . . . . . . . . 15 |- ( a C_ On -> ( x e. a -> x e. On ) )
12	9, 10, 11	syl2im 39	. . . . . . . . . . . . . 14 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. On ) )
13		eloni 5432	. . . . . . . . . . . . . 14 |- ( x e. On -> Ord x )
14	12, 13	syl6 34	. . . . . . . . . . . . 13 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> Ord x ) )
15		ordtr 5436	. . . . . . . . . . . . 13 |- ( Ord x -> Tr x )
16	14, 15	syl6 34	. . . . . . . . . . . 12 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> Tr x ) )
17		simpll 759	. . . . . . . . . . . . . 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 3652	. . . . . . . . . . . . . 14 |- ( a i^i x ) C_ x
20	19	sseli 3427	. . . . . . . . . . . . 13 |- ( y e. ( a i^i x ) -> y e. x )
21	18, 20	syl8 72	. . . . . . . . . . . 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 4503	. . . . . . . . . . . . 13 |- ( Tr x -> ( ( z e. y /\ y e. x ) -> z e. x ) )
23	22	expcomd 440	. . . . . . . . . . . 12 |- ( Tr x -> ( y e. x -> ( z e. y -> z e. x ) ) )
24	16, 21, 5, 23	ee233 36870	. . . . . . . . . . 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 3616	. . . . . . . . . . . 12 |- ( z e. ( a i^i x ) <-> ( z e. a /\ z e. x ) )
26	25	simplbi2 630	. . . . . . . . . . 11 |- ( z e. a -> ( z e. x -> z e. ( a i^i x ) ) )
27	8, 24, 26	ee33 36872	. . . . . . . . . 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 3616	. . . . . . . . . . 11 |- ( z e. ( ( a i^i x ) i^i y ) <-> ( z e. ( a i^i x ) /\ z e. y ) )
29	28	simplbi2com 632	. . . . . . . . . 10 |- ( z e. y -> ( z e. ( a i^i x ) -> z e. ( ( a i^i x ) i^i y ) ) )
30	5, 27, 29	ee33 36872	. . . . . . . . 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 610	. . . . . . . 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 36905	. . . . . . 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 3420	. . . . . . . 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 210	. . . . . . 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 72	. . . . . 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 463	. . . . . . 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 3765	. . . . . . 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 436	. . . . . 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 36872	. . . . 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 459	. . . . . . 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 3651	. . . . . . 7 |- ( a i^i x ) C_ a
44	43	sseli 3427	. . . . . 6 |- ( y e. ( a i^i x ) -> y e. a )
45	42, 44	syl8 72	. . . . 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		pm3.21 450	. . . . 5 |- ( ( a i^i y ) = (/) -> ( y e. a -> ( y e. a /\ ( a i^i y ) = (/) ) ) )
47	40, 45, 46	ee33 36872	. . . 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 1774	. . 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 36904	. . . 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 2742	. . . 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 230	. . 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 1705	. . 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 66	. 2 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> E. y ( y e. a /\ ( a i^i y ) = (/) ) ) )
54		df-rex 2742	. 2 |- ( E. y e. a ( a i^i y ) = (/) <-> E. y ( y e. a /\ ( a i^i y ) = (/) ) )
55	53, 54	syl6ibr 231	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 371   A.wal 1441   = wceq 1443   E.wex 1662   e. wcel 1886   =/= wne 2621   E.wrex 2737   i^i cin 3402   C_ wss 3403   (/)c0 3730   Tr wtr 4496   Ord word 5421   Oncon0 5422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-tr 4497  df-eprel 4744  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-ord 5425  df-on 5426

This theorem is referenced by:  onfrALT  36909