Theorem extex 5916

Description: The class of all extensional relationships is a set. (Contributed by SF, 19-Feb-2015.)

Assertion

Ref	Expression
extex	|- Ext e. _V

Proof of Theorem extex

Dummy variables p a r x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ext 5908	. . 3 |- Ext = {<.r, a>. | A.x e. a A.y e. a (A.z e. a (zrx <-> zry) -> x = y)}
2		vex 2863	. . . . . . 7 |- r e. _V
3		vex 2863	. . . . . . 7 |- a e. _V
4	2, 3	opex 4589	. . . . . 6 |- <.r, a>. e. _V
5	4	elcompl 3226	. . . . 5 |- (<.r, a>. e. ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c) <-> -. <.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c))
6		elin 3220	. . . . . . . . . 10 |- (<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c)) <-> (<.{x}, <.r, a>.>. e. Ins2 S /\ <.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c)))
7	2	otelins2 5792	. . . . . . . . . . . 12 |- (<.{x}, <.r, a>.>. e. Ins2 S <-> <.{x}, a>. e. S )
8		vex 2863	. . . . . . . . . . . . 13 |- x e. _V
9	8, 3	opelssetsn 4761	. . . . . . . . . . . 12 |- (<.{x}, a>. e. S <-> x e. a)
10	7, 9	bitri 240	. . . . . . . . . . 11 |- (<.{x}, <.r, a>.>. e. Ins2 S <-> x e. a)
11		elin 3220	. . . . . . . . . . . . . . 15 |- (<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I )) <-> (<.{y}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S /\ <.{y}, <.{x}, <.r, a>.>.>. e. ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I )))
12		snex 4112	. . . . . . . . . . . . . . . . . 18 |- {x} e. _V
13	12	otelins2 5792	. . . . . . . . . . . . . . . . 17 |- (<.{y}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S <-> <.{y}, <.r, a>.>. e. Ins2 S )
14	2	otelins2 5792	. . . . . . . . . . . . . . . . 17 |- (<.{y}, <.r, a>.>. e. Ins2 S <-> <.{y}, a>. e. S )
15		vex 2863	. . . . . . . . . . . . . . . . . 18 |- y e. _V
16	15, 3	opelssetsn 4761	. . . . . . . . . . . . . . . . 17 |- (<.{y}, a>. e. S <-> y e. a)
17	13, 14, 16	3bitri 262	. . . . . . . . . . . . . . . 16 |- (<.{y}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S <-> y e. a)
18		eldif 3222	. . . . . . . . . . . . . . . . 17 |- (<.{y}, <.{x}, <.r, a>.>.>. e. ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ) <-> (<.{y}, <.{x}, <.r, a>.>.>. e. ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) /\ -. <.{y}, <.{x}, <.r, a>.>.>. e. Ins3 _I ))
19		snex 4112	. . . . . . . . . . . . . . . . . . . . . 22 |- {y} e. _V
20	12, 4	opex 4589	. . . . . . . . . . . . . . . . . . . . . 22 |- <.{x}, <.r, a>.>. e. _V
21	19, 20	opex 4589	. . . . . . . . . . . . . . . . . . . . 21 |- <.{y}, <.{x}, <.r, a>.>.>. e. _V
22	21	elcompl 3226	. . . . . . . . . . . . . . . . . . . 20 |- (<.{y}, <.{x}, <.r, a>.>.>. e. ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) <-> -. <.{y}, <.{x}, <.r, a>.>.>. e. (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c))
23		elin 3220	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. ( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c))) <-> (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins2 Ins2 S /\ <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c))))
24	19	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins2 Ins2 S <-> <.{z}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S )
25	12	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{z}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S <-> <.{z}, <.r, a>.>. e. Ins2 S )
26	2	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.{z}, <.r, a>.>. e. Ins2 S <-> <.{z}, a>. e. S )
27		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- z e. _V
28	27, 3	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.{z}, a>. e. S <-> z e. a)
29	26, 28	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{z}, <.r, a>.>. e. Ins2 S <-> z e. a)
30	24, 25, 29	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins2 Ins2 S <-> z e. a)
31		elsymdif 3224	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)) <-> -. (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))
32	19	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> <.{z}, <.{x}, <.r, a>.>.>. e. Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c))
33		elima1c 4948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (<.{z}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> E.p<.{p}, <.{z}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ))
34		elin 3220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (<.{p}, <.{z}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ) <-> (<.{p}, <.{z}, <.{x}, r>.>.>. e. Ins4 SI3 _I /\ <.{p}, <.{z}, <.{x}, r>.>.>. e. Ins2 Ins2 S ))
35	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (<.{p}, <.{z}, <.{x}, r>.>.>. e. Ins4 SI3 _I <-> <.{p}, <.{z}, {x}>.>. e. SI3 _I )
36		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- p e. _V
37	36, 27, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (<.{p}, <.{z}, {x}>.>. e. SI3 _I <-> <.p, <.z, x>.>. e. _I )
38		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (p _I <.z, x>. <-> <.p, <.z, x>.>. e. _I )
39	27, 8	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- <.z, x>. e. _V
40	39	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (p _I <.z, x>. <-> p = <.z, x>.)
41	37, 38, 40	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (<.{p}, <.{z}, {x}>.>. e. SI3 _I <-> p = <.z, x>.)
42	35, 41	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (<.{p}, <.{z}, <.{x}, r>.>.>. e. Ins4 SI3 _I <-> p = <.z, x>.)
43		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- {z} e. _V
44	43	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (<.{p}, <.{z}, <.{x}, r>.>.>. e. Ins2 Ins2 S <-> <.{p}, <.{x}, r>.>. e. Ins2 S )
45	12	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (<.{p}, <.{x}, r>.>. e. Ins2 S <-> <.{p}, r>. e. S )
46	36, 2	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (<.{p}, r>. e. S <-> p e. r)
47	44, 45, 46	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (<.{p}, <.{z}, <.{x}, r>.>.>. e. Ins2 Ins2 S <-> p e. r)
48	42, 47	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((<.{p}, <.{z}, <.{x}, r>.>.>. e. Ins4 SI3 _I /\ <.{p}, <.{z}, <.{x}, r>.>.>. e. Ins2 Ins2 S ) <-> (p = <.z, x>. /\ p e. r))
49	34, 48	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (<.{p}, <.{z}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ) <-> (p = <.z, x>. /\ p e. r))
50	49	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (E.p<.{p}, <.{z}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ) <-> E.p(p = <.z, x>. /\ p e. r))
51	33, 50	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (<.{z}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> E.p(p = <.z, x>. /\ p e. r))
52	3	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (<.{z}, <.{x}, <.r, a>.>.>. e. Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> <.{z}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c))
53		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (zrx <-> <.z, x>. e. r)
54		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (<.z, x>. e. r <-> E.p(p = <.z, x>. /\ p e. r))
55	53, 54	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (zrx <-> E.p(p = <.z, x>. /\ p e. r))
56	51, 52, 55	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (<.{z}, <.{x}, <.r, a>.>.>. e. Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> zrx)
57	32, 56	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> zrx)
58		elin 3220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S ) <-> (<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. Ins4 SI3 _I /\ <.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. Ins2 Ins2 Ins2 Ins3 S ))
59	20	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. Ins4 SI3 _I <-> <.{p}, <.{z}, {y}>.>. e. SI3 _I )
60	36, 27, 15	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (<.{p}, <.{z}, {y}>.>. e. SI3 _I <-> <.p, <.z, y>.>. e. _I )
61		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (p _I <.z, y>. <-> <.p, <.z, y>.>. e. _I )
62	27, 15	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- <.z, y>. e. _V
63	62	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (p _I <.z, y>. <-> p = <.z, y>.)
64	60, 61, 63	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (<.{p}, <.{z}, {y}>.>. e. SI3 _I <-> p = <.z, y>.)
65	59, 64	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. Ins4 SI3 _I <-> p = <.z, y>.)
66	43	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. Ins2 Ins2 Ins2 Ins3 S <-> <.{p}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins2 Ins3 S )
67	19	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (<.{p}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins2 Ins3 S <-> <.{p}, <.{x}, <.r, a>.>.>. e. Ins2 Ins3 S )
68	12	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (<.{p}, <.{x}, <.r, a>.>.>. e. Ins2 Ins3 S <-> <.{p}, <.r, a>.>. e. Ins3 S )
69	3	otelins3 5793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (<.{p}, <.r, a>.>. e. Ins3 S <-> <.{p}, r>. e. S )
70	68, 69, 46	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (<.{p}, <.{x}, <.r, a>.>.>. e. Ins2 Ins3 S <-> p e. r)
71	66, 67, 70	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. Ins2 Ins2 Ins2 Ins3 S <-> p e. r)
72	65, 71	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. Ins4 SI3 _I /\ <.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. Ins2 Ins2 Ins2 Ins3 S ) <-> (p = <.z, y>. /\ p e. r))
73	58, 72	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S ) <-> (p = <.z, y>. /\ p e. r))
74	73	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (E.p<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S ) <-> E.p(p = <.z, y>. /\ p e. r))
75		elima1c 4948	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c) <-> E.p<.{p}, <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S ))
76		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (zry <-> <.z, y>. e. r)
77		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (<.z, y>. e. r <-> E.p(p = <.z, y>. /\ p e. r))
78	76, 77	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (zry <-> E.p(p = <.z, y>. /\ p e. r))
79	74, 75, 78	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c) <-> zry)
80	57, 79	bibi12i 306	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)) <-> (zrx <-> zry))
81	31, 80	xchbinx 301	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)) <-> -. (zrx <-> zry))
82	30, 81	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. Ins2 Ins2 Ins2 S /\ <.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c))) <-> (z e. a /\ -. (zrx <-> zry)))
83	23, 82	bitri 240	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. ( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c))) <-> (z e. a /\ -. (zrx <-> zry)))
84	83	exbii 1582	. . . . . . . . . . . . . . . . . . . . 21 |- (E.z<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. ( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c))) <-> E.z(z e. a /\ -. (zrx <-> zry)))
85		elima1c 4948	. . . . . . . . . . . . . . . . . . . . 21 |- (<.{y}, <.{x}, <.r, a>.>.>. e. (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) <-> E.z<.{z}, <.{y}, <.{x}, <.r, a>.>.>.>. e. ( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c))))
86		df-rex 2621	. . . . . . . . . . . . . . . . . . . . 21 |- (E.z e. a -. (zrx <-> zry) <-> E.z(z e. a /\ -. (zrx <-> zry)))
87	84, 85, 86	3bitr4i 268	. . . . . . . . . . . . . . . . . . . 20 |- (<.{y}, <.{x}, <.r, a>.>.>. e. (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) <-> E.z e. a -. (zrx <-> zry))
88	22, 87	xchbinx 301	. . . . . . . . . . . . . . . . . . 19 |- (<.{y}, <.{x}, <.r, a>.>.>. e. ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) <-> -. E.z e. a -. (zrx <-> zry))
89		dfral2 2627	. . . . . . . . . . . . . . . . . . 19 |- (A.z e. a (zrx <-> zry) <-> -. E.z e. a -. (zrx <-> zry))
90	88, 89	bitr4i 243	. . . . . . . . . . . . . . . . . 18 |- (<.{y}, <.{x}, <.r, a>.>.>. e. ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) <-> A.z e. a (zrx <-> zry))
91	4	otelins3 5793	. . . . . . . . . . . . . . . . . . . 20 |- (<.{y}, <.{x}, <.r, a>.>.>. e. Ins3 _I <-> <.{y}, {x}>. e. _I )
92		df-br 4641	. . . . . . . . . . . . . . . . . . . 20 |- ({y} _I {x} <-> <.{y}, {x}>. e. _I )
93	12	ideq 4871	. . . . . . . . . . . . . . . . . . . . 21 |- ({y} _I {x} <-> {y} = {x})
94	15	sneqb 3877	. . . . . . . . . . . . . . . . . . . . 21 |- ({y} = {x} <-> y = x)
95		equcom 1680	. . . . . . . . . . . . . . . . . . . . 21 |- (y = x <-> x = y)
96	93, 94, 95	3bitri 262	. . . . . . . . . . . . . . . . . . . 20 |- ({y} _I {x} <-> x = y)
97	91, 92, 96	3bitr2i 264	. . . . . . . . . . . . . . . . . . 19 |- (<.{y}, <.{x}, <.r, a>.>.>. e. Ins3 _I <-> x = y)
98	97	notbii 287	. . . . . . . . . . . . . . . . . 18 |- (-. <.{y}, <.{x}, <.r, a>.>.>. e. Ins3 _I <-> -. x = y)
99	90, 98	anbi12i 678	. . . . . . . . . . . . . . . . 17 |- ((<.{y}, <.{x}, <.r, a>.>.>. e. ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) /\ -. <.{y}, <.{x}, <.r, a>.>.>. e. Ins3 _I ) <-> (A.z e. a (zrx <-> zry) /\ -. x = y))
100	18, 99	bitri 240	. . . . . . . . . . . . . . . 16 |- (<.{y}, <.{x}, <.r, a>.>.>. e. ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ) <-> (A.z e. a (zrx <-> zry) /\ -. x = y))
101	17, 100	anbi12i 678	. . . . . . . . . . . . . . 15 |- ((<.{y}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S /\ <.{y}, <.{x}, <.r, a>.>.>. e. ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I )) <-> (y e. a /\ (A.z e. a (zrx <-> zry) /\ -. x = y)))
102	11, 101	bitri 240	. . . . . . . . . . . . . 14 |- (<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I )) <-> (y e. a /\ (A.z e. a (zrx <-> zry) /\ -. x = y)))
103	102	exbii 1582	. . . . . . . . . . . . 13 |- (E.y<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I )) <-> E.y(y e. a /\ (A.z e. a (zrx <-> zry) /\ -. x = y)))
104		elima1c 4948	. . . . . . . . . . . . 13 |- (<.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c) <-> E.y<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I )))
105		df-rex 2621	. . . . . . . . . . . . 13 |- (E.y e. a (A.z e. a (zrx <-> zry) /\ -. x = y) <-> E.y(y e. a /\ (A.z e. a (zrx <-> zry) /\ -. x = y)))
106	103, 104, 105	3bitr4i 268	. . . . . . . . . . . 12 |- (<.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c) <-> E.y e. a (A.z e. a (zrx <-> zry) /\ -. x = y))
107		rexanali 2661	. . . . . . . . . . . 12 |- (E.y e. a (A.z e. a (zrx <-> zry) /\ -. x = y) <-> -. A.y e. a (A.z e. a (zrx <-> zry) -> x = y))
108	106, 107	bitri 240	. . . . . . . . . . 11 |- (<.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c) <-> -. A.y e. a (A.z e. a (zrx <-> zry) -> x = y))
109	10, 108	anbi12i 678	. . . . . . . . . 10 |- ((<.{x}, <.r, a>.>. e. Ins2 S /\ <.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c)) <-> (x e. a /\ -. A.y e. a (A.z e. a (zrx <-> zry) -> x = y)))
110	6, 109	bitri 240	. . . . . . . . 9 |- (<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c)) <-> (x e. a /\ -. A.y e. a (A.z e. a (zrx <-> zry) -> x = y)))
111	110	exbii 1582	. . . . . . . 8 |- (E.x<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c)) <-> E.x(x e. a /\ -. A.y e. a (A.z e. a (zrx <-> zry) -> x = y)))
112		elima1c 4948	. . . . . . . 8 |- (<.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c) <-> E.x<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c)))
113		df-rex 2621	. . . . . . . 8 |- (E.x e. a -. A.y e. a (A.z e. a (zrx <-> zry) -> x = y) <-> E.x(x e. a /\ -. A.y e. a (A.z e. a (zrx <-> zry) -> x = y)))
114	111, 112, 113	3bitr4i 268	. . . . . . 7 |- (<.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c) <-> E.x e. a -. A.y e. a (A.z e. a (zrx <-> zry) -> x = y))
115		rexnal 2626	. . . . . . 7 |- (E.x e. a -. A.y e. a (A.z e. a (zrx <-> zry) -> x = y) <-> -. A.x e. a A.y e. a (A.z e. a (zrx <-> zry) -> x = y))
116	114, 115	bitri 240	. . . . . 6 |- (<.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c) <-> -. A.x e. a A.y e. a (A.z e. a (zrx <-> zry) -> x = y))
117	116	con2bii 322	. . . . 5 |- (A.x e. a A.y e. a (A.z e. a (zrx <-> zry) -> x = y) <-> -. <.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c))
118	5, 117	bitr4i 243	. . . 4 |- (<.r, a>. e. ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c) <-> A.x e. a A.y e. a (A.z e. a (zrx <-> zry) -> x = y))
119	118	opabbi2i 4867	. . 3 |- ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c) = {<.r, a>. | A.x e. a A.y e. a (A.z e. a (zrx <-> zry) -> x = y)}
120	1, 119	eqtr4i 2376	. 2 |- Ext = ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c)
121		ssetex 4745	. . . . . 6 |- S e. _V
122	121	ins2ex 5798	. . . . 5 |- Ins2 S e. _V
123	122	ins2ex 5798	. . . . . . 7 |- Ins2 Ins2 S e. _V
124	123	ins2ex 5798	. . . . . . . . . . 11 |- Ins2 Ins2 Ins2 S e. _V
125		idex 5505	. . . . . . . . . . . . . . . . . 18 |- _I e. _V
126	125	si3ex 5807	. . . . . . . . . . . . . . . . 17 |- SI3 _I e. _V
127	126	ins4ex 5800	. . . . . . . . . . . . . . . 16 |- Ins4 SI3 _I e. _V
128	127, 123	inex 4106	. . . . . . . . . . . . . . 15 |- ( Ins4 SI3 _I i^i Ins2 Ins2 S ) e. _V
129		1cex 4143	. . . . . . . . . . . . . . 15 |- 1c e. _V
130	128, 129	imaex 4748	. . . . . . . . . . . . . 14 |- (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) e. _V
131	130	ins4ex 5800	. . . . . . . . . . . . 13 |- Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) e. _V
132	131	ins2ex 5798	. . . . . . . . . . . 12 |- Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) e. _V
133	121	ins3ex 5799	. . . . . . . . . . . . . . . . 17 |- Ins3 S e. _V
134	133	ins2ex 5798	. . . . . . . . . . . . . . . 16 |- Ins2 Ins3 S e. _V
135	134	ins2ex 5798	. . . . . . . . . . . . . . 15 |- Ins2 Ins2 Ins3 S e. _V
136	135	ins2ex 5798	. . . . . . . . . . . . . 14 |- Ins2 Ins2 Ins2 Ins3 S e. _V
137	127, 136	inex 4106	. . . . . . . . . . . . 13 |- ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S ) e. _V
138	137, 129	imaex 4748	. . . . . . . . . . . 12 |- (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c) e. _V
139	132, 138	symdifex 4109	. . . . . . . . . . 11 |- ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)) e. _V
140	124, 139	inex 4106	. . . . . . . . . 10 |- ( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c))) e. _V
141	140, 129	imaex 4748	. . . . . . . . 9 |- (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) e. _V
142	141	complex 4105	. . . . . . . 8 |- ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) e. _V
143	125	ins3ex 5799	. . . . . . . 8 |- Ins3 _I e. _V
144	142, 143	difex 4108	. . . . . . 7 |- ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ) e. _V
145	123, 144	inex 4106	. . . . . 6 |- ( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I )) e. _V
146	145, 129	imaex 4748	. . . . 5 |- (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c) e. _V
147	122, 146	inex 4106	. . . 4 |- ( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c)) e. _V
148	147, 129	imaex 4748	. . 3 |- (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c) e. _V
149	148	complex 4105	. 2 |- ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i ( ∼ (( Ins2 Ins2 Ins2 S i^i ( Ins2 Ins4 (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) (+) (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 Ins3 S )"1c)))"1c) \ Ins3 _I ))"1c))"1c) e. _V
150	120, 149	eqeltri 2423	1 |- Ext e. _V

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  A.wral 2615  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   \ cdif 3207   i^i cin 3209   (+) csymdif 3210  {csn 3738  1_cc1c 4135  <.cop 4562  {copab 4623   class class class wbr 4640   S csset 4720  "cima 4723   _I cid 4764   Ins2 cins2 5750   Ins3 cins3 5752   Ins4 cins4 5756   SI₃ csi3 5758   Ext cext 5897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-cnv 4786  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759  df-ext 5908

This theorem is referenced by: (None)