Theorem symex 5917

Description: The class of all symmetric relationships is a set. (Contributed by SF, 20-Feb-2015.)

Assertion

Ref	Expression
symex	|- Sym e. _V

Proof of Theorem symex

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

Step	Hyp	Ref	Expression
1		df-sym 5909	. . 3 |- Sym = {<.r, a>. | A.x e. a A.y e. a (xry -> yrx)}
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 Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c) <-> -. <.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c))
6		elin 3220	. . . . . . . . . 10 |- (<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c)) <-> (<.{x}, <.r, a>.>. e. Ins2 S /\ <.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"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 Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c))) <-> (<.{y}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S /\ <.{y}, <.{x}, <.r, a>.>.>. e. Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c))))
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	3	oqelins4 5795	. . . . . . . . . . . . . . . . 17 |- (<.{y}, <.{x}, <.r, a>.>.>. e. Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) <-> <.{y}, <.{x}, r>.>. e. ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))
19		eldif 3222	. . . . . . . . . . . . . . . . 17 |- (<.{y}, <.{x}, r>.>. e. ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) <-> (<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) /\ -. <.{y}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))
20		elin 3220	. . . . . . . . . . . . . . . . . . . . 21 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 Swap i^i Ins2 Ins2 S ) <-> (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 Swap /\ <.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S ))
21	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 Swap <-> <.{p}, <.{y}, {x}>.>. e. SI3 Swap )
22		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- p e. _V
23	22, 15, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{p}, <.{y}, {x}>.>. e. SI3 Swap <-> <.p, <.y, x>.>. e. Swap )
24		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (p Swap <.y, x>. <-> <.p, <.y, x>.>. e. Swap )
25	15, 8	brswap2 4861	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (p Swap <.y, x>. <-> p = <.x, y>.)
26	23, 24, 25	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{p}, <.{y}, {x}>.>. e. SI3 Swap <-> p = <.x, y>.)
27	21, 26	bitri 240	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 Swap <-> p = <.x, y>.)
28		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . 24 |- {y} e. _V
29	28	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S <-> <.{p}, <.{x}, r>.>. e. Ins2 S )
30	12	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{p}, <.{x}, r>.>. e. Ins2 S <-> <.{p}, r>. e. S )
31	22, 2	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{p}, r>. e. S <-> p e. r)
32	29, 30, 31	3bitri 262	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S <-> p e. r)
33	27, 32	anbi12i 678	. . . . . . . . . . . . . . . . . . . . 21 |- ((<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 Swap /\ <.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S ) <-> (p = <.x, y>. /\ p e. r))
34	20, 33	bitri 240	. . . . . . . . . . . . . . . . . . . 20 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 Swap i^i Ins2 Ins2 S ) <-> (p = <.x, y>. /\ p e. r))
35	34	exbii 1582	. . . . . . . . . . . . . . . . . . 19 |- (E.p<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 Swap i^i Ins2 Ins2 S ) <-> E.p(p = <.x, y>. /\ p e. r))
36		elima1c 4948	. . . . . . . . . . . . . . . . . . 19 |- (<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) <-> E.p<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 Swap i^i Ins2 Ins2 S ))
37		df-br 4641	. . . . . . . . . . . . . . . . . . . 20 |- (xry <-> <.x, y>. e. r)
38		df-clel 2349	. . . . . . . . . . . . . . . . . . . 20 |- (<.x, y>. e. r <-> E.p(p = <.x, y>. /\ p e. r))
39	37, 38	bitri 240	. . . . . . . . . . . . . . . . . . 19 |- (xry <-> E.p(p = <.x, y>. /\ p e. r))
40	35, 36, 39	3bitr4i 268	. . . . . . . . . . . . . . . . . 18 |- (<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) <-> xry)
41		elin 3220	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ) <-> (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 _I /\ <.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S ))
42	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 _I <-> <.{p}, <.{y}, {x}>.>. e. SI3 _I )
43	22, 15, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{p}, <.{y}, {x}>.>. e. SI3 _I <-> <.p, <.y, x>.>. e. _I )
44		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (p _I <.y, x>. <-> <.p, <.y, x>.>. e. _I )
45	15, 8	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- <.y, x>. e. _V
46	45	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (p _I <.y, x>. <-> p = <.y, x>.)
47	43, 44, 46	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{p}, <.{y}, {x}>.>. e. SI3 _I <-> p = <.y, x>.)
48	42, 47	bitri 240	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 _I <-> p = <.y, x>.)
49	48, 32	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . 22 |- ((<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 _I /\ <.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S ) <-> (p = <.y, x>. /\ p e. r))
50	41, 49	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ) <-> (p = <.y, x>. /\ p e. r))
51	50	exbii 1582	. . . . . . . . . . . . . . . . . . . 20 |- (E.p<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ) <-> E.p(p = <.y, x>. /\ p e. r))
52		elima1c 4948	. . . . . . . . . . . . . . . . . . . 20 |- (<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> E.p<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ))
53		df-br 4641	. . . . . . . . . . . . . . . . . . . . 21 |- (yrx <-> <.y, x>. e. r)
54		df-clel 2349	. . . . . . . . . . . . . . . . . . . . 21 |- (<.y, x>. e. r <-> E.p(p = <.y, x>. /\ p e. r))
55	53, 54	bitri 240	. . . . . . . . . . . . . . . . . . . 20 |- (yrx <-> E.p(p = <.y, x>. /\ p e. r))
56	51, 52, 55	3bitr4i 268	. . . . . . . . . . . . . . . . . . 19 |- (<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> yrx)
57	56	notbii 287	. . . . . . . . . . . . . . . . . 18 |- (-. <.{y}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> -. yrx)
58	40, 57	anbi12i 678	. . . . . . . . . . . . . . . . 17 |- ((<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) /\ -. <.{y}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) <-> (xry /\ -. yrx))
59	18, 19, 58	3bitri 262	. . . . . . . . . . . . . . . 16 |- (<.{y}, <.{x}, <.r, a>.>.>. e. Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) <-> (xry /\ -. yrx))
60	17, 59	anbi12i 678	. . . . . . . . . . . . . . 15 |- ((<.{y}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S /\ <.{y}, <.{x}, <.r, a>.>.>. e. Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c))) <-> (y e. a /\ (xry /\ -. yrx)))
61	11, 60	bitri 240	. . . . . . . . . . . . . 14 |- (<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c))) <-> (y e. a /\ (xry /\ -. yrx)))
62	61	exbii 1582	. . . . . . . . . . . . 13 |- (E.y<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c))) <-> E.y(y e. a /\ (xry /\ -. yrx)))
63		elima1c 4948	. . . . . . . . . . . . 13 |- (<.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c) <-> E.y<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c))))
64		df-rex 2621	. . . . . . . . . . . . 13 |- (E.y e. a (xry /\ -. yrx) <-> E.y(y e. a /\ (xry /\ -. yrx)))
65	62, 63, 64	3bitr4i 268	. . . . . . . . . . . 12 |- (<.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c) <-> E.y e. a (xry /\ -. yrx))
66		rexanali 2661	. . . . . . . . . . . 12 |- (E.y e. a (xry /\ -. yrx) <-> -. A.y e. a (xry -> yrx))
67	65, 66	bitri 240	. . . . . . . . . . 11 |- (<.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c) <-> -. A.y e. a (xry -> yrx))
68	10, 67	anbi12i 678	. . . . . . . . . 10 |- ((<.{x}, <.r, a>.>. e. Ins2 S /\ <.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c)) <-> (x e. a /\ -. A.y e. a (xry -> yrx)))
69	6, 68	bitri 240	. . . . . . . . 9 |- (<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c)) <-> (x e. a /\ -. A.y e. a (xry -> yrx)))
70	69	exbii 1582	. . . . . . . 8 |- (E.x<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c)) <-> E.x(x e. a /\ -. A.y e. a (xry -> yrx)))
71		elima1c 4948	. . . . . . . 8 |- (<.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c) <-> E.x<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c)))
72		df-rex 2621	. . . . . . . 8 |- (E.x e. a -. A.y e. a (xry -> yrx) <-> E.x(x e. a /\ -. A.y e. a (xry -> yrx)))
73	70, 71, 72	3bitr4i 268	. . . . . . 7 |- (<.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c) <-> E.x e. a -. A.y e. a (xry -> yrx))
74		rexnal 2626	. . . . . . 7 |- (E.x e. a -. A.y e. a (xry -> yrx) <-> -. A.x e. a A.y e. a (xry -> yrx))
75	73, 74	bitri 240	. . . . . 6 |- (<.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c) <-> -. A.x e. a A.y e. a (xry -> yrx))
76	75	con2bii 322	. . . . 5 |- (A.x e. a A.y e. a (xry -> yrx) <-> -. <.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c))
77	5, 76	bitr4i 243	. . . 4 |- (<.r, a>. e. ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c) <-> A.x e. a A.y e. a (xry -> yrx))
78	77	opabbi2i 4867	. . 3 |- ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c) = {<.r, a>. | A.x e. a A.y e. a (xry -> yrx)}
79	1, 78	eqtr4i 2376	. 2 |- Sym = ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c)
80		ssetex 4745	. . . . . 6 |- S e. _V
81	80	ins2ex 5798	. . . . 5 |- Ins2 S e. _V
82	81	ins2ex 5798	. . . . . . 7 |- Ins2 Ins2 S e. _V
83		swapex 4743	. . . . . . . . . . . . 13 |- Swap e. _V
84	83	si3ex 5807	. . . . . . . . . . . 12 |- SI3 Swap e. _V
85	84	ins4ex 5800	. . . . . . . . . . 11 |- Ins4 SI3 Swap e. _V
86	85, 82	inex 4106	. . . . . . . . . 10 |- ( Ins4 SI3 Swap i^i Ins2 Ins2 S ) e. _V
87		1cex 4143	. . . . . . . . . 10 |- 1c e. _V
88	86, 87	imaex 4748	. . . . . . . . 9 |- (( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) e. _V
89		idex 5505	. . . . . . . . . . . . 13 |- _I e. _V
90	89	si3ex 5807	. . . . . . . . . . . 12 |- SI3 _I e. _V
91	90	ins4ex 5800	. . . . . . . . . . 11 |- Ins4 SI3 _I e. _V
92	91, 82	inex 4106	. . . . . . . . . 10 |- ( Ins4 SI3 _I i^i Ins2 Ins2 S ) e. _V
93	92, 87	imaex 4748	. . . . . . . . 9 |- (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) e. _V
94	88, 93	difex 4108	. . . . . . . 8 |- ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) e. _V
95	94	ins4ex 5800	. . . . . . 7 |- Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) e. _V
96	82, 95	inex 4106	. . . . . 6 |- ( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c))) e. _V
97	96, 87	imaex 4748	. . . . 5 |- (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c) e. _V
98	81, 97	inex 4106	. . . 4 |- ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c)) e. _V
99	98, 87	imaex 4748	. . 3 |- (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c) e. _V
100	99	complex 4105	. 2 |- ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 ((( Ins4 SI3 Swap i^i Ins2 Ins2 S )"1c) \ (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))"1c))"1c) e. _V
101	79, 100	eqeltri 2423	1 |- Sym e. _V

Syntax hints:  -. wn 3   -> wi 4   /\ 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  {csn 3738  1_cc1c 4135  <.cop 4562  {copab 4623   class class class wbr 4640   Swap cswap 4719   S csset 4720  "cima 4723   _I cid 4764   Ins2 cins2 5750   Ins4 cins4 5756   SI₃ csi3 5758   Sym csym 5898

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-ins4 5757  df-si3 5759  df-sym 5909

This theorem is referenced by:  erex  5921