Theorem onfrALTlem2VD 39125

Description: Virtual deduction proof of onfrALTlem2 38761. 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. onfrALTlem2 38761 is onfrALTlem2VD 39125 without virtual deductions and was automatically derived from onfrALTlem2VD 39125.

1::	|- (. ( 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 ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) ).
2:1:	|- (. ( 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 ) ).
3:2:	|- (. ( 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 ).
4::	|- (. ( a C_ On /\ a =/= (/) ) ->. ( a C_ On /\ a =/= (/) ) ).
5::	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. ( x e. a /\ -. ( a i^i x ) = (/) ) ).
6:5:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. x e. a ).
7:4:	|- (. ( a C_ On /\ a =/= (/) ) ->. a C_ On ).
8:6,7:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. x e. On ).
9:8:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. Ord x ).
10:9:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. Tr x ).
11:1:	|- (. ( 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 ) ).
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 ) ) ->. y e. x ).
13:2:	|- (. ( 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 ).
14:10,12,13:	|- (. ( 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 ) ) ->. z e. x ).
15:3,14:	|- (. ( 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 ) ).
16:13,15:	|- (. ( 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 ) ).
17:16:	|- (. ( 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 ) ) ).
18:17:	|- (. ( 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 ) ) ).
19:18:	|- (. ( 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 ) ).
20::	|- (. ( 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 ) /\ ( ( a i^i x ) i^i y ) = (/) ) ).
21:20:	|- (. ( 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 ) = (/) ).
22:19,21:	|- (. ( 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 ) = (/) ).
23:20:	|- (. ( 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 ) ).
24:23:	|- (. ( 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 ).
25:22,24:	|- (. ( 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 ) = (/) ) ).
26:25:	|- (. ( 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 ) = (/) ) ) ).
27:26:	|- (. ( 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 ) = (/) ) ) ).
28:27:	|- (. ( 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 ) = (/) ) -> E. y ( y e. a /\ ( a i^i y ) = (/) ) ) ).
29::	|- (. ( 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 ) = (/) ).
30:29:	|- (. ( 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 ) = (/) ) ).
31:28,30:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. E. y ( y e. a /\ ( a i^i y ) = (/) ) ).
qed:31:	|- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. E. y e. a ( a i^i y ) = (/) ).

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

Assertion

Ref	Expression
onfrALTlem2VD	|- (. ( 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 onfrALTlem2VD

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn3 38840	. . . . . . . . . . . . . 14 |- (. ( 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 ) /\ ( ( a i^i x ) i^i y ) = (/) ) /\ z e. ( a i^i y ) ) ).
2		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( 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	1, 2	e3 38964	. . . . . . . . . . . . 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 ) ) ->. z e. ( a i^i y ) ).
4		inss2 3834	. . . . . . . . . . . . . 14 |- ( a i^i y ) C_ y
5	4	sseli 3599	. . . . . . . . . . . . 13 |- ( z e. ( a i^i y ) -> z e. y )
6	3, 5	e3 38964	. . . . . . . . . . . 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 ) ) ->. z e. y ).
7		inss1 3833	. . . . . . . . . . . . . . 15 |- ( a i^i y ) C_ a
8	7	sseli 3599	. . . . . . . . . . . . . 14 |- ( z e. ( a i^i y ) -> z e. a )
9	3, 8	e3 38964	. . . . . . . . . . . . 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 ) ) ->. z e. a ).
10		idn2 38838	. . . . . . . . . . . . . . . . . 18 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. ( x e. a /\ -. ( a i^i x ) = (/) ) ).
11		simpl 473	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. a )
12	10, 11	e2 38856	. . . . . . . . . . . . . . . . 17 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. x e. a ).
13		idn1 38790	. . . . . . . . . . . . . . . . . 18 |- (. ( a C_ On /\ a =/= (/) ) ->. ( a C_ On /\ a =/= (/) ) ).
14		simpl 473	. . . . . . . . . . . . . . . . . 18 |- ( ( a C_ On /\ a =/= (/) ) -> a C_ On )
15	13, 14	e1a 38852	. . . . . . . . . . . . . . . . 17 |- (. ( a C_ On /\ a =/= (/) ) ->. a C_ On ).
16		ssel 3597	. . . . . . . . . . . . . . . . . 18 |- ( a C_ On -> ( x e. a -> x e. On ) )
17	16	com12 32	. . . . . . . . . . . . . . . . 17 |- ( x e. a -> ( a C_ On -> x e. On ) )
18	12, 15, 17	e21 38957	. . . . . . . . . . . . . . . 16 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. x e. On ).
19		eloni 5733	. . . . . . . . . . . . . . . 16 |- ( x e. On -> Ord x )
20	18, 19	e2 38856	. . . . . . . . . . . . . . 15 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. Ord x ).
21		ordtr 5737	. . . . . . . . . . . . . . 15 |- ( Ord x -> Tr x )
22	20, 21	e2 38856	. . . . . . . . . . . . . 14 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. Tr x ).
23		simpll 790	. . . . . . . . . . . . . . . 16 |- ( ( ( 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 ) )
24	1, 23	e3 38964	. . . . . . . . . . . . . . 15 |- (. ( 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 ) ).
25		inss2 3834	. . . . . . . . . . . . . . . 16 |- ( a i^i x ) C_ x
26	25	sseli 3599	. . . . . . . . . . . . . . 15 |- ( y e. ( a i^i x ) -> y e. x )
27	24, 26	e3 38964	. . . . . . . . . . . . . 14 |- (. ( 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 ).
28		trel 4759	. . . . . . . . . . . . . . 15 |- ( Tr x -> ( ( z e. y /\ y e. x ) -> z e. x ) )
29	28	expcomd 454	. . . . . . . . . . . . . 14 |- ( Tr x -> ( y e. x -> ( z e. y -> z e. x ) ) )
30	22, 27, 6, 29	e233 38992	. . . . . . . . . . . . 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 ) ) ->. z e. x ).
31		elin 3796	. . . . . . . . . . . . . 14 |- ( z e. ( a i^i x ) <-> ( z e. a /\ z e. x ) )
32	31	simplbi2 655	. . . . . . . . . . . . 13 |- ( z e. a -> ( z e. x -> z e. ( a i^i x ) ) )
33	9, 30, 32	e33 38961	. . . . . . . . . . . 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 ) ) ->. z e. ( a i^i x ) ).
34		elin 3796	. . . . . . . . . . . . 13 |- ( z e. ( ( a i^i x ) i^i y ) <-> ( z e. ( a i^i x ) /\ z e. y ) )
35	34	simplbi2com 657	. . . . . . . . . . . 12 |- ( z e. y -> ( z e. ( a i^i x ) -> z e. ( ( a i^i x ) i^i y ) ) )
36	6, 33, 35	e33 38961	. . . . . . . . . . 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 x ) i^i y ) ).
37	36	in3an 38836	. . . . . . . . . 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 ) i^i y ) ) ).
38	37	gen31 38846	. . . . . . . . 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 ) = (/) ) ->. A. z ( z e. ( a i^i y ) -> z e. ( ( a i^i x ) i^i y ) ) ).
39		dfss2 3591	. . . . . . . . . 10 |- ( ( 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 ) ) )
40	39	biimpri 218	. . . . . . . . 9 |- ( 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 ) )
41	38, 40	e3 38964	. . . . . . . 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 ) = (/) ) ->. ( a i^i y ) C_ ( ( a i^i x ) i^i y ) ).
42		idn3 38840	. . . . . . . . 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 ) = (/) ) ->. ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) ).
43		simpr 477	. . . . . . . . 9 |- ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( ( a i^i x ) i^i y ) = (/) )
44	42, 43	e3 38964	. . . . . . . 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 ) = (/) ) ->. ( ( a i^i x ) i^i y ) = (/) ).
45		sseq0 3975	. . . . . . . . 9 |- ( ( ( a i^i y ) C_ ( ( a i^i x ) i^i y ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> ( a i^i y ) = (/) )
46	45	ex 450	. . . . . . . 8 |- ( ( a i^i y ) C_ ( ( a i^i x ) i^i y ) -> ( ( ( a i^i x ) i^i y ) = (/) -> ( a i^i y ) = (/) ) )
47	41, 44, 46	e33 38961	. . . . . . 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 i^i y ) = (/) ).
48		simpl 473	. . . . . . . . 9 |- ( ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> y e. ( a i^i x ) )
49	42, 48	e3 38964	. . . . . . . 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 ) = (/) ) ->. y e. ( a i^i x ) ).
50		inss1 3833	. . . . . . . . 9 |- ( a i^i x ) C_ a
51	50	sseli 3599	. . . . . . . 8 |- ( y e. ( a i^i x ) -> y e. a )
52	49, 51	e3 38964	. . . . . . 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 ) = (/) ) ->. y e. a ).
53		pm3.21 464	. . . . . . 7 |- ( ( a i^i y ) = (/) -> ( y e. a -> ( y e. a /\ ( a i^i y ) = (/) ) ) )
54	47, 52, 53	e33 38961	. . . . . 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 /\ ( a i^i y ) = (/) ) ).
55	54	in3 38834	. . . . 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 /\ ( a i^i y ) = (/) ) ) ).
56	55	gen21 38844	. . . 4 |- (. ( 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 ) = (/) ) ) ).
57		exim 1761	. . . 4 |- ( 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 ) = (/) ) ) )
58	56, 57	e2 38856	. . 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 ) = (/) ) -> E. y ( y e. a /\ ( a i^i y ) = (/) ) ) ).
59		onfrALTlem3VD 39123	. . . 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 ) = (/) ).
60		df-rex 2918	. . . . 5 |- ( 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 ) = (/) ) )
61	60	biimpi 206	. . . 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 ) = (/) ) )
62	59, 61	e2 38856	. . 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 ) = (/) ) ).
63		id 22	. . 3 |- ( ( E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) -> E. 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 ) = (/) ) ) )
64	58, 62, 63	e22 38896	. 2 |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. E. y ( y e. a /\ ( a i^i y ) = (/) ) ).
65		df-rex 2918	. . 3 |- ( E. y e. a ( a i^i y ) = (/) <-> E. y ( y e. a /\ ( a i^i y ) = (/) ) )
66	65	biimpri 218	. 2 |- ( E. y ( y e. a /\ ( a i^i y ) = (/) ) -> E. y e. a ( a i^i y ) = (/) )
67	64, 66	e2 38856	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   (.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  df-vd3 38806

This theorem is referenced by:  onfrALTVD  39127