Theorem onfrALTlem3 38759

Description: Lemma for onfrALT 38764. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALTlem3	|- ( ( 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 ) = (/) ) )

Distinct variable groups:   y, a   x, y

Proof of Theorem onfrALTlem3

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3624	. . 3 |- ( a i^i x ) C_ ( a i^i x )
2		simpr 477	. . . . 5 |- ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> -. ( a i^i x ) = (/) )
3	2	a1i 11	. . . 4 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> -. ( a i^i x ) = (/) ) )
4		df-ne 2795	. . . 4 |- ( ( a i^i x ) =/= (/) <-> -. ( a i^i x ) = (/) )
5	3, 4	syl6ibr 242	. . 3 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( a i^i x ) =/= (/) ) )
6		pm3.2 463	. . 3 |- ( ( a i^i x ) C_ ( a i^i x ) -> ( ( a i^i x ) =/= (/) -> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) ) )
7	1, 5, 6	mpsylsyld 69	. 2 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) ) )
8		vex 3203	. . . . 5 |- x e. _V
9	8	inex2 4800	. . . 4 |- ( a i^i x ) e. _V
10		inss2 3834	. . . . . . 7 |- ( a i^i x ) C_ x
11		simpl 473	. . . . . . . . . 10 |- ( ( a C_ On /\ a =/= (/) ) -> a C_ On )
12		simpl 473	. . . . . . . . . 10 |- ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. a )
13		ssel 3597	. . . . . . . . . 10 |- ( a C_ On -> ( x e. a -> x e. On ) )
14	11, 12, 13	syl2im 40	. . . . . . . . 9 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. On ) )
15		eloni 5733	. . . . . . . . 9 |- ( x e. On -> Ord x )
16	14, 15	syl6 35	. . . . . . . 8 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> Ord x ) )
17		ordwe 5736	. . . . . . . 8 |- ( Ord x -> _E We x )
18	16, 17	syl6 35	. . . . . . 7 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> _E We x ) )
19		wess 5101	. . . . . . 7 |- ( ( a i^i x ) C_ x -> ( _E We x -> _E We ( a i^i x ) ) )
20	10, 18, 19	mpsylsyld 69	. . . . . 6 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> _E We ( a i^i x ) ) )
21		wefr 5104	. . . . . 6 |- ( _E We ( a i^i x ) -> _E Fr ( a i^i x ) )
22	20, 21	syl6 35	. . . . 5 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> _E Fr ( a i^i x ) ) )
23		dfepfr 5099	. . . . 5 |- ( _E Fr ( a i^i x ) <-> A. b ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) )
24	22, 23	syl6ib 241	. . . 4 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> A. b ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) ) )
25		spsbc 3448	. . . 4 |- ( ( a i^i x ) e. _V -> ( A. b ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) -> [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) ) )
26	9, 24, 25	mpsylsyld 69	. . 3 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) ) )
27		onfrALTlem5 38757	. . 3 |- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )
28	26, 27	syl6ib 241	. 2 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) ) )
29	7, 28	mpdd 43	1 |- ( ( 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 ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   C_ wss 3574   (/)c0 3915   _E cep 5028   Fr wfr 5070   We wwe 5072   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:  onfrALTlem2  38761