Theorem onfrALTlem3VD 39123

Description: Virtual deduction proof of onfrALTlem3 38759. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem3 38759 is onfrALTlem3VD 39123 without virtual deductions and was automatically derived from onfrALTlem3VD 39123.

1::	|- (. ( a C_ On /\ a =/= (/) ) ->. ( a C_ On /\ a =/= (/) ) ).
2::	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. ( x e. a /\ -. ( a i^i x ) = (/) ) ).
3:2:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. x e. a ).
4:1:	|- (. ( a C_ On /\ a =/= (/) ) ->. a C_ On ).
5:3,4:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. x e. On ).
6:5:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. Ord x ).
7:6:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. _E We x ).
8::	|- ( a i^i x ) C_ x
9:7,8:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. _E We ( a i^i x ) ).
10:9:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. _E Fr ( a i^i x ) ).
11:10:	|- (. ( 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 ) = (/) ) ).
12::	|- x e. _V
13:12,8:	|- ( a i^i x ) e. _V
14:13,11:	|- (. ( 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 ) = (/) ) ).
15::	|- ( [. ( 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 ) = (/) ) )
16:14,15:	|- (. ( 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 ) = (/) ) ).
17::	|- ( a i^i x ) C_ ( a i^i x )
18:2:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. -. ( a i^i x ) = (/) ).
19:18:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. ( a i^i x ) =/= (/) ).
20:17,19:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) ).
qed:16,20:	|- (. ( 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 ) = (/) ).

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALTlem3VD	|- (. ( 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 onfrALTlem3VD

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- x e. _V
2		inss2 3834	. . . . 5 |- ( a i^i x ) C_ x
3	1, 2	ssexi 4803	. . . 4 |- ( a i^i x ) e. _V
4		idn2 38838	. . . . . . . . . . 11 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. ( x e. a /\ -. ( a i^i x ) = (/) ) ).
5		simpl 473	. . . . . . . . . . 11 |- ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. a )
6	4, 5	e2 38856	. . . . . . . . . 10 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. x e. a ).
7		idn1 38790	. . . . . . . . . . 11 |- (. ( a C_ On /\ a =/= (/) ) ->. ( a C_ On /\ a =/= (/) ) ).
8		simpl 473	. . . . . . . . . . 11 |- ( ( a C_ On /\ a =/= (/) ) -> a C_ On )
9	7, 8	e1a 38852	. . . . . . . . . 10 |- (. ( a C_ On /\ a =/= (/) ) ->. a C_ On ).
10		ssel 3597	. . . . . . . . . . 11 |- ( a C_ On -> ( x e. a -> x e. On ) )
11	10	com12 32	. . . . . . . . . 10 |- ( x e. a -> ( a C_ On -> x e. On ) )
12	6, 9, 11	e21 38957	. . . . . . . . 9 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. x e. On ).
13		eloni 5733	. . . . . . . . 9 |- ( x e. On -> Ord x )
14	12, 13	e2 38856	. . . . . . . 8 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. Ord x ).
15		ordwe 5736	. . . . . . . 8 |- ( Ord x -> _E We x )
16	14, 15	e2 38856	. . . . . . 7 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. _E We x ).
17		wess 5101	. . . . . . . 8 |- ( ( a i^i x ) C_ x -> ( _E We x -> _E We ( a i^i x ) ) )
18	17	com12 32	. . . . . . 7 |- ( _E We x -> ( ( a i^i x ) C_ x -> _E We ( a i^i x ) ) )
19	16, 2, 18	e20 38954	. . . . . 6 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. _E We ( a i^i x ) ).
20		wefr 5104	. . . . . 6 |- ( _E We ( a i^i x ) -> _E Fr ( a i^i x ) )
21	19, 20	e2 38856	. . . . 5 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. _E Fr ( a i^i x ) ).
22		dfepfr 5099	. . . . . 6 |- ( _E Fr ( a i^i x ) <-> A. b ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) )
23	22	biimpi 206	. . . . 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	21, 23	e2 38856	. . . 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	3, 24, 25	e02 38922	. . 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	e2bi 38857	. 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		ssid 3624	. . 3 |- ( a i^i x ) C_ ( a i^i x )
30		simpr 477	. . . . 5 |- ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> -. ( a i^i x ) = (/) )
31	4, 30	e2 38856	. . . 4 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. -. ( a i^i x ) = (/) ).
32		df-ne 2795	. . . . 5 |- ( ( a i^i x ) =/= (/) <-> -. ( a i^i x ) = (/) )
33	32	biimpri 218	. . . 4 |- ( -. ( a i^i x ) = (/) -> ( a i^i x ) =/= (/) )
34	31, 33	e2 38856	. . 3 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. ( a i^i x ) =/= (/) ).
35		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 ) =/= (/) ) ) )
36	29, 34, 35	e02 38922	. 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 ) =/= (/) ) ).
37		id 22	. 2 |- ( ( ( ( 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 ) = (/) ) -> ( ( ( 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 ) = (/) ) )
38	28, 36, 37	e22 38896	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   (.wvd2 38793

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  df-vd1 38786  df-vd2 38794

This theorem is referenced by:  onfrALTlem2VD  39125