Theorem symex 5917

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

Assertion

Ref	Expression
symex	⊢ Sym ∈ 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⟩ ∣ ∀x ∈ a ∀y ∈ a (xry → yrx)}
2		vex 2863	. . . . . . 7 ⊢ r ∈ V
3		vex 2863	. . . . . . 7 ⊢ a ∈ V
4	2, 3	opex 4589	. . . . . 6 ⊢ ⟨r, a⟩ ∈ V
5	4	elcompl 3226	. . . . 5 ⊢ (⟨r, a⟩ ∈ ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) ↔ ¬ ⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c))
6		elin 3220	. . . . . . . . . 10 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) ↔ (⟨{x}, ⟨r, a⟩⟩ ∈ Ins2 S ∧ ⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)))
7	2	otelins2 5792	. . . . . . . . . . . 12 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ Ins2 S ↔ ⟨{x}, a⟩ ∈ S )
8		vex 2863	. . . . . . . . . . . . 13 ⊢ x ∈ V
9	8, 3	opelssetsn 4761	. . . . . . . . . . . 12 ⊢ (⟨{x}, a⟩ ∈ S ↔ x ∈ a)
10	7, 9	bitri 240	. . . . . . . . . . 11 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ Ins2 S ↔ x ∈ a)
11		elin 3220	. . . . . . . . . . . . . . 15 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) ↔ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins2 Ins2 S ∧ ⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))))
12		snex 4112	. . . . . . . . . . . . . . . . . 18 ⊢ {x} ∈ V
13	12	otelins2 5792	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{y}, ⟨r, a⟩⟩ ∈ Ins2 S )
14	2	otelins2 5792	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, ⟨r, a⟩⟩ ∈ Ins2 S ↔ ⟨{y}, a⟩ ∈ S )
15		vex 2863	. . . . . . . . . . . . . . . . . 18 ⊢ y ∈ V
16	15, 3	opelssetsn 4761	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, a⟩ ∈ S ↔ y ∈ a)
17	13, 14, 16	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins2 Ins2 S ↔ y ∈ a)
18	3	oqelins4 5795	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ ⟨{y}, ⟨{x}, r⟩⟩ ∈ ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)))
19		eldif 3222	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∧ ¬ ⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)))
20		elin 3220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ↔ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 Swap ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ))
21	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 Swap ↔ ⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 Swap )
22		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ p ∈ V
23	22, 15, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 Swap ↔ ⟨p, ⟨y, x⟩⟩ ∈ Swap )
24		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (p Swap ⟨y, x⟩ ↔ ⟨p, ⟨y, x⟩⟩ ∈ Swap )
25	15, 8	brswap2 4861	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (p Swap ⟨y, x⟩ ↔ p = ⟨x, y⟩)
26	23, 24, 25	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 Swap ↔ p = ⟨x, y⟩)
27	21, 26	bitri 240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 Swap ↔ p = ⟨x, y⟩)
28		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {y} ∈ V
29	28	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{p}, ⟨{x}, r⟩⟩ ∈ Ins2 S )
30	12	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{x}, r⟩⟩ ∈ Ins2 S ↔ ⟨{p}, r⟩ ∈ S )
31	22, 2	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, r⟩ ∈ S ↔ p ∈ r)
32	29, 30, 31	3bitri 262	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ↔ p ∈ r)
33	27, 32	anbi12i 678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 Swap ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ) ↔ (p = ⟨x, y⟩ ∧ p ∈ r))
34	20, 33	bitri 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ↔ (p = ⟨x, y⟩ ∧ p ∈ r))
35	34	exbii 1582	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ↔ ∃p(p = ⟨x, y⟩ ∧ p ∈ r))
36		elima1c 4948	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ↔ ∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ))
37		df-br 4641	. . . . . . . . . . . . . . . . . . . 20 ⊢ (xry ↔ ⟨x, y⟩ ∈ r)
38		df-clel 2349	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨x, y⟩ ∈ r ↔ ∃p(p = ⟨x, y⟩ ∧ p ∈ r))
39	37, 38	bitri 240	. . . . . . . . . . . . . . . . . . 19 ⊢ (xry ↔ ∃p(p = ⟨x, y⟩ ∧ p ∈ r))
40	35, 36, 39	3bitr4i 268	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ↔ xry)
41		elin 3220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ))
42	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ↔ ⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 I )
43	22, 15, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 I ↔ ⟨p, ⟨y, x⟩⟩ ∈ I )
44		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (p I ⟨y, x⟩ ↔ ⟨p, ⟨y, x⟩⟩ ∈ I )
45	15, 8	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ⟨y, x⟩ ∈ V
46	45	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (p I ⟨y, x⟩ ↔ p = ⟨y, x⟩)
47	43, 44, 46	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 I ↔ p = ⟨y, x⟩)
48	42, 47	bitri 240	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ↔ p = ⟨y, x⟩)
49	48, 32	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ) ↔ (p = ⟨y, x⟩ ∧ p ∈ r))
50	41, 49	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ (p = ⟨y, x⟩ ∧ p ∈ r))
51	50	exbii 1582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ ∃p(p = ⟨y, x⟩ ∧ p ∈ r))
52		elima1c 4948	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ ∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ))
53		df-br 4641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (yrx ↔ ⟨y, x⟩ ∈ r)
54		df-clel 2349	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨y, x⟩ ∈ r ↔ ∃p(p = ⟨y, x⟩ ∧ p ∈ r))
55	53, 54	bitri 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (yrx ↔ ∃p(p = ⟨y, x⟩ ∧ p ∈ r))
56	51, 52, 55	3bitr4i 268	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ yrx)
57	56	notbii 287	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ ¬ yrx)
58	40, 57	anbi12i 678	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∧ ¬ ⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (xry ∧ ¬ yrx))
59	18, 19, 58	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (xry ∧ ¬ yrx))
60	17, 59	anbi12i 678	. . . . . . . . . . . . . . 15 ⊢ ((⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins2 Ins2 S ∧ ⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) ↔ (y ∈ a ∧ (xry ∧ ¬ yrx)))
61	11, 60	bitri 240	. . . . . . . . . . . . . 14 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) ↔ (y ∈ a ∧ (xry ∧ ¬ yrx)))
62	61	exbii 1582	. . . . . . . . . . . . 13 ⊢ (∃y⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) ↔ ∃y(y ∈ a ∧ (xry ∧ ¬ yrx)))
63		elima1c 4948	. . . . . . . . . . . . 13 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c) ↔ ∃y⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))))
64		df-rex 2621	. . . . . . . . . . . . 13 ⊢ (∃y ∈ a (xry ∧ ¬ yrx) ↔ ∃y(y ∈ a ∧ (xry ∧ ¬ yrx)))
65	62, 63, 64	3bitr4i 268	. . . . . . . . . . . 12 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c) ↔ ∃y ∈ a (xry ∧ ¬ yrx))
66		rexanali 2661	. . . . . . . . . . . 12 ⊢ (∃y ∈ a (xry ∧ ¬ yrx) ↔ ¬ ∀y ∈ a (xry → yrx))
67	65, 66	bitri 240	. . . . . . . . . . 11 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c) ↔ ¬ ∀y ∈ a (xry → yrx))
68	10, 67	anbi12i 678	. . . . . . . . . 10 ⊢ ((⟨{x}, ⟨r, a⟩⟩ ∈ Ins2 S ∧ ⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) ↔ (x ∈ a ∧ ¬ ∀y ∈ a (xry → yrx)))
69	6, 68	bitri 240	. . . . . . . . 9 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) ↔ (x ∈ a ∧ ¬ ∀y ∈ a (xry → yrx)))
70	69	exbii 1582	. . . . . . . 8 ⊢ (∃x⟨{x}, ⟨r, a⟩⟩ ∈ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) ↔ ∃x(x ∈ a ∧ ¬ ∀y ∈ a (xry → yrx)))
71		elima1c 4948	. . . . . . . 8 ⊢ (⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) ↔ ∃x⟨{x}, ⟨r, a⟩⟩ ∈ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)))
72		df-rex 2621	. . . . . . . 8 ⊢ (∃x ∈ a ¬ ∀y ∈ a (xry → yrx) ↔ ∃x(x ∈ a ∧ ¬ ∀y ∈ a (xry → yrx)))
73	70, 71, 72	3bitr4i 268	. . . . . . 7 ⊢ (⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) ↔ ∃x ∈ a ¬ ∀y ∈ a (xry → yrx))
74		rexnal 2626	. . . . . . 7 ⊢ (∃x ∈ a ¬ ∀y ∈ a (xry → yrx) ↔ ¬ ∀x ∈ a ∀y ∈ a (xry → yrx))
75	73, 74	bitri 240	. . . . . 6 ⊢ (⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) ↔ ¬ ∀x ∈ a ∀y ∈ a (xry → yrx))
76	75	con2bii 322	. . . . 5 ⊢ (∀x ∈ a ∀y ∈ a (xry → yrx) ↔ ¬ ⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c))
77	5, 76	bitr4i 243	. . . 4 ⊢ (⟨r, a⟩ ∈ ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) ↔ ∀x ∈ a ∀y ∈ a (xry → yrx))
78	77	opabbi2i 4867	. . 3 ⊢ ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) = {⟨r, a⟩ ∣ ∀x ∈ a ∀y ∈ a (xry → yrx)}
79	1, 78	eqtr4i 2376	. 2 ⊢ Sym = ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c)
80		ssetex 4745	. . . . . 6 ⊢ S ∈ V
81	80	ins2ex 5798	. . . . 5 ⊢ Ins2 S ∈ V
82	81	ins2ex 5798	. . . . . . 7 ⊢ Ins2 Ins2 S ∈ V
83		swapex 4743	. . . . . . . . . . . . 13 ⊢ Swap ∈ V
84	83	si3ex 5807	. . . . . . . . . . . 12 ⊢ SI3 Swap ∈ V
85	84	ins4ex 5800	. . . . . . . . . . 11 ⊢ Ins4 SI3 Swap ∈ V
86	85, 82	inex 4106	. . . . . . . . . 10 ⊢ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ∈ V
87		1cex 4143	. . . . . . . . . 10 ⊢ 1c ∈ V
88	86, 87	imaex 4748	. . . . . . . . 9 ⊢ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∈ V
89		idex 5505	. . . . . . . . . . . . 13 ⊢ I ∈ V
90	89	si3ex 5807	. . . . . . . . . . . 12 ⊢ SI3 I ∈ V
91	90	ins4ex 5800	. . . . . . . . . . 11 ⊢ Ins4 SI3 I ∈ V
92	91, 82	inex 4106	. . . . . . . . . 10 ⊢ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ∈ V
93	92, 87	imaex 4748	. . . . . . . . 9 ⊢ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ∈ V
94	88, 93	difex 4108	. . . . . . . 8 ⊢ ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∈ V
95	94	ins4ex 5800	. . . . . . 7 ⊢ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∈ V
96	82, 95	inex 4106	. . . . . 6 ⊢ ( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) ∈ V
97	96, 87	imaex 4748	. . . . 5 ⊢ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c) ∈ V
98	81, 97	inex 4106	. . . 4 ⊢ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) ∈ V
99	98, 87	imaex 4748	. . 3 ⊢ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) ∈ V
100	99	complex 4105	. 2 ⊢ ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∖ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) ∈ V
101	79, 100	eqeltri 2423	1 ⊢ Sym ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∩ 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