Theorem onfrALTlem5VD 37282

Description: Virtual deduction proof of onfrALTlem5 36908. 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. onfrALTlem5 36908 is onfrALTlem5VD 37282 without virtual deductions and was automatically derived from onfrALTlem5VD 37282.

1::	|- a e. _V
2:1:	|- ( a i^i x ) e. _V
3:2:	|- ( [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) = (/) )
4:3:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> -. ( a i^i x ) = (/) )
5::	|- ( ( a i^i x ) =/= (/) <-> -. ( a i^i x ) = (/) )
6:4,5:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) =/= (/) )
7:2:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
8::	|- ( b =/= (/) <-> -. b = (/) )
9:8:	|- A. b ( b =/= (/) <-> -. b = (/) )
10:2,9:	|- ( [. ( a i^i x ) / b ]. b =/= (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
11:7,10:	|- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. b =/= (/) )
12:6,11:	|- ( [. ( a i^i x ) / b ]. b =/= (/) <-> ( a i^i x ) =/= (/) )
13:2:	|- ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) )
14:12,13:	|- ( ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
15:2:	|- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) )
16:15,14:	|- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
17:2:	|- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y )
18:2:	|- [_ ( a i^i x ) / b ]_ b = ( a i^i x )
19:2:	|- [_ ( a i^i x ) / b ]_ y = y
20:18,19:	|- ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y ) = ( ( a i^i x ) i^i y )
21:17,20:	|- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( ( a i^i x ) i^i y )
22:2:	|- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) )
23:2:	|- [_ ( a i^i x ) / b ]_ (/) = (/)
24:21,23:	|- ( [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) <-> ( ( a i^i x ) i^i y ) = (/) )
25:22,24:	|- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> ( ( a i^i x ) i^i y ) = (/) )
26:2:	|- ( [. ( a i^i x ) / b ]. y e. b <-> y e. ( a i^i x ) )
27:25,26:	|- ( ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
28:2:	|- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) )
29:27,28:	|- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
30:29:	|- A. y ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
31:30:	|- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
32::	|- ( 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 ) = (/) ) )
33:31,32:	|- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
34:2:	|- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
35:33,34:	|- ( [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
36::	|- ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
37:36:	|- A. b ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
38:2,37:	|- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
39:35,38:	|- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
40:16,39:	|- ( ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( 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 ) = (/) ) )
41:2:	|- ( [. ( a i^i x ) / 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 =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) )
qed:40,41:	|- ( [. ( 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 ) = (/) ) )

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

Assertion

Ref	Expression
onfrALTlem5VD	|- ( [. ( 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 ) = (/) ) )

Distinct variable groups:   a, b, y   x, b, y

Proof of Theorem onfrALTlem5VD

Step	Hyp	Ref	Expression
1		vex 3048	. . . 4 |- a e. _V
2	1	inex1 4544	. . 3 |- ( a i^i x ) e. _V
3		sbcimg 3309	. . 3 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / 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 =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) ) )
4	2, 3	e0a 37159	. 2 |- ( [. ( a i^i x ) / 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 =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) )
5		sbcangOLD 36890	. . . . 5 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) ) )
6	2, 5	e0a 37159	. . . 4 |- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) )
7		sseq1 3453	. . . . . . 7 |- ( b = ( a i^i x ) -> ( b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) ) )
8	7	sbcieg 3300	. . . . . 6 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) ) )
9	2, 8	e0a 37159	. . . . 5 |- ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) )
10		sbcng 3308	. . . . . . . . 9 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. -. b = (/) <-> -. [. ( a i^i x ) / b ]. b = (/) ) )
11	10	bicomd 205	. . . . . . . 8 |- ( ( a i^i x ) e. _V -> ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. -. b = (/) ) )
12	2, 11	e0a 37159	. . . . . . 7 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
13		df-ne 2624	. . . . . . . . 9 |- ( b =/= (/) <-> -. b = (/) )
14	13	ax-gen 1669	. . . . . . . 8 |- A. b ( b =/= (/) <-> -. b = (/) )
15		sbcbi 36900	. . . . . . . 8 |- ( ( a i^i x ) e. _V -> ( A. b ( b =/= (/) <-> -. b = (/) ) -> ( [. ( a i^i x ) / b ]. b =/= (/) <-> [. ( a i^i x ) / b ]. -. b = (/) ) ) )
16	2, 14, 15	e00 37155	. . . . . . 7 |- ( [. ( a i^i x ) / b ]. b =/= (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
17	12, 16	bitr4i 256	. . . . . 6 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. b =/= (/) )
18		eqsbc3 3307	. . . . . . . . 9 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) = (/) ) )
19	2, 18	e0a 37159	. . . . . . . 8 |- ( [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) = (/) )
20	19	notbii 298	. . . . . . 7 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> -. ( a i^i x ) = (/) )
21		df-ne 2624	. . . . . . 7 |- ( ( a i^i x ) =/= (/) <-> -. ( a i^i x ) = (/) )
22	20, 21	bitr4i 256	. . . . . 6 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) =/= (/) )
23	17, 22	bitr3i 255	. . . . 5 |- ( [. ( a i^i x ) / b ]. b =/= (/) <-> ( a i^i x ) =/= (/) )
24	9, 23	anbi12i 703	. . . 4 |- ( ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
25	6, 24	bitri 253	. . 3 |- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
26		df-rex 2743	. . . . . 6 |- ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
27	26	ax-gen 1669	. . . . 5 |- A. b ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
28		sbcbi 36900	. . . . 5 |- ( ( a i^i x ) e. _V -> ( A. b ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) ) -> ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) ) ) )
29	2, 27, 28	e00 37155	. . . 4 |- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
30		sbcexgOLD 36904	. . . . . . 7 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) ) )
31	30	bicomd 205	. . . . . 6 |- ( ( a i^i x ) e. _V -> ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) ) )
32	2, 31	e0a 37159	. . . . 5 |- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
33		sbcangOLD 36890	. . . . . . . . . 10 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) ) )
34	2, 33	e0a 37159	. . . . . . . . 9 |- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) )
35		sbcel2gv 3327	. . . . . . . . . . 11 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. y e. b <-> y e. ( a i^i x ) ) )
36	2, 35	e0a 37159	. . . . . . . . . 10 |- ( [. ( a i^i x ) / b ]. y e. b <-> y e. ( a i^i x ) )
37		sbceqg 3773	. . . . . . . . . . . 12 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) ) )
38	2, 37	e0a 37159	. . . . . . . . . . 11 |- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) )
39		csbingOLD 37215	. . . . . . . . . . . . . 14 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y ) )
40	2, 39	e0a 37159	. . . . . . . . . . . . 13 |- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y )
41		csbvarg 3792	. . . . . . . . . . . . . . 15 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ b = ( a i^i x ) )
42	2, 41	e0a 37159	. . . . . . . . . . . . . 14 |- [_ ( a i^i x ) / b ]_ b = ( a i^i x )
43		csbconstg 3376	. . . . . . . . . . . . . . 15 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ y = y )
44	2, 43	e0a 37159	. . . . . . . . . . . . . 14 |- [_ ( a i^i x ) / b ]_ y = y
45	42, 44	ineq12i 3632	. . . . . . . . . . . . 13 |- ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y ) = ( ( a i^i x ) i^i y )
46	40, 45	eqtri 2473	. . . . . . . . . . . 12 |- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( ( a i^i x ) i^i y )
47		csbconstg 3376	. . . . . . . . . . . . 13 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ (/) = (/) )
48	2, 47	e0a 37159	. . . . . . . . . . . 12 |- [_ ( a i^i x ) / b ]_ (/) = (/)
49	46, 48	eqeq12i 2465	. . . . . . . . . . 11 |- ( [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) <-> ( ( a i^i x ) i^i y ) = (/) )
50	38, 49	bitri 253	. . . . . . . . . 10 |- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> ( ( a i^i x ) i^i y ) = (/) )
51	36, 50	anbi12i 703	. . . . . . . . 9 |- ( ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
52	34, 51	bitri 253	. . . . . . . 8 |- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
53	52	ax-gen 1669	. . . . . . 7 |- A. y ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
54		exbi 1716	. . . . . . 7 |- ( A. y ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) ) -> ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) ) )
55	53, 54	e0a 37159	. . . . . 6 |- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
56		df-rex 2743	. . . . . 6 |- ( 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 ) = (/) ) )
57	55, 56	bitr4i 256	. . . . 5 |- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
58	32, 57	bitr3i 255	. . . 4 |- ( [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
59	29, 58	bitri 253	. . 3 |- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
60	25, 59	imbi12i 328	. 2 |- ( ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( 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 ) = (/) ) )
61	4, 60	bitri 253	1 |- ( [. ( 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 ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1442   = wceq 1444   E.wex 1663   e. wcel 1887   =/= wne 2622   E.wrex 2738   _Vcvv 3045   [.wsbc 3267   [_csb 3363   i^i cin 3403   C_ wss 3404   (/)c0 3731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-in 3411  df-ss 3418

This theorem is referenced by: (None)