Theorem connexex 5914

Description: The class of all connected relationships is a set. (Contributed by SF, 11-Mar-2015.)

Assertion

Ref	Expression
connexex	⊢ Connex ∈ V

Proof of Theorem connexex

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

Step	Hyp	Ref	Expression
1		df-connex 5904	. . 3 ⊢ Connex = {⟨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		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)))
7		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)))
8	2	otelins2 5792	. . . . . . . . . . . 12 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ Ins2 S ↔ ⟨{x}, a⟩ ∈ S )
9		vex 2863	. . . . . . . . . . . . 13 ⊢ x ∈ V
10	9, 3	opelssetsn 4761	. . . . . . . . . . . 12 ⊢ (⟨{x}, a⟩ ∈ S ↔ x ∈ a)
11	8, 10	bitri 240	. . . . . . . . . . 11 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ Ins2 S ↔ x ∈ a)
12		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))))
13		eldif 3222	. . . . . . . . . . . . . . 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))))
14		snex 4112	. . . . . . . . . . . . . . . . . 18 ⊢ {x} ∈ V
15	14	otelins2 5792	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{y}, ⟨r, a⟩⟩ ∈ Ins2 S )
16	2	otelins2 5792	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, ⟨r, a⟩⟩ ∈ Ins2 S ↔ ⟨{y}, a⟩ ∈ S )
17		vex 2863	. . . . . . . . . . . . . . . . . 18 ⊢ y ∈ V
18	17, 3	opelssetsn 4761	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, a⟩ ∈ S ↔ y ∈ a)
19	15, 16, 18	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins2 Ins2 S ↔ y ∈ a)
20	3	oqelins4 5795	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨{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)))
21		elun 3221	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨{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)))
22		elin 3220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ↔ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 Swap ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ))
23	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 Swap ↔ ⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 Swap )
24		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ p ∈ V
25	24, 17, 9	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 Swap ↔ ⟨p, ⟨y, x⟩⟩ ∈ Swap )
26		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (p Swap ⟨y, x⟩ ↔ ⟨p, ⟨y, x⟩⟩ ∈ Swap )
27	17, 9	brswap2 4861	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (p Swap ⟨y, x⟩ ↔ p = ⟨x, y⟩)
28	26, 27	bitr3i 242	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨p, ⟨y, x⟩⟩ ∈ Swap ↔ p = ⟨x, y⟩)
29	23, 25, 28	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 Swap ↔ p = ⟨x, y⟩)
30		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {y} ∈ V
31	30	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{p}, ⟨{x}, r⟩⟩ ∈ Ins2 S )
32	14	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{x}, r⟩⟩ ∈ Ins2 S ↔ ⟨{p}, r⟩ ∈ S )
33	24, 2	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, r⟩ ∈ S ↔ p ∈ r)
34	31, 32, 33	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ↔ p ∈ r)
35	29, 34	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 Swap ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ) ↔ (p = ⟨x, y⟩ ∧ p ∈ r))
36	22, 35	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ↔ (p = ⟨x, y⟩ ∧ p ∈ r))
37	36	exbii 1582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) ↔ ∃p(p = ⟨x, y⟩ ∧ p ∈ r))
38		elima1c 4948	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ↔ ∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 Swap ∩ Ins2 Ins2 S ))
39		df-br 4641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (xry ↔ ⟨x, y⟩ ∈ r)
40		df-clel 2349	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨x, y⟩ ∈ r ↔ ∃p(p = ⟨x, y⟩ ∧ p ∈ r))
41	39, 40	bitri 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (xry ↔ ∃p(p = ⟨x, y⟩ ∧ p ∈ r))
42	37, 38, 41	3bitr4i 268	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ↔ xry)
43		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 ))
44	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ↔ ⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 I )
45	24, 17, 9	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 I ↔ ⟨p, ⟨y, x⟩⟩ ∈ I )
46		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (p I ⟨y, x⟩ ↔ ⟨p, ⟨y, x⟩⟩ ∈ I )
47	17, 9	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ⟨y, x⟩ ∈ V
48	47	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (p I ⟨y, x⟩ ↔ p = ⟨y, x⟩)
49	46, 48	bitr3i 242	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨p, ⟨y, x⟩⟩ ∈ I ↔ p = ⟨y, x⟩)
50	44, 45, 49	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ↔ p = ⟨y, x⟩)
51	50, 34	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ) ↔ (p = ⟨y, x⟩ ∧ p ∈ r))
52	43, 51	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ (p = ⟨y, x⟩ ∧ p ∈ r))
53	52	exbii 1582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ ∃p(p = ⟨y, x⟩ ∧ p ∈ r))
54		elima1c 4948	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ ∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ))
55		df-br 4641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (yrx ↔ ⟨y, x⟩ ∈ r)
56		df-clel 2349	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨y, x⟩ ∈ r ↔ ∃p(p = ⟨y, x⟩ ∧ p ∈ r))
57	55, 56	bitri 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (yrx ↔ ∃p(p = ⟨y, x⟩ ∧ p ∈ r))
58	53, 54, 57	3bitr4i 268	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ yrx)
59	42, 58	orbi12i 507	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∨ ⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (xry ∨ yrx))
60	20, 21, 59	3bitri 262	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (xry ∨ yrx))
61	60	notbii 287	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ ¬ (xry ∨ yrx))
62	19, 61	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)))
63	13, 62	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)))
64	63	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)))
65	12, 64	bitri 240	. . . . . . . . . . . 12 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∖ Ins4 ((( Ins4 SI3 Swap ∩ Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c) ↔ ∃y(y ∈ a ∧ ¬ (xry ∨ yrx)))
66		df-rex 2621	. . . . . . . . . . . 12 ⊢ (∃y ∈ a ¬ (xry ∨ yrx) ↔ ∃y(y ∈ a ∧ ¬ (xry ∨ yrx)))
67		rexnal 2626	. . . . . . . . . . . 12 ⊢ (∃y ∈ a ¬ (xry ∨ yrx) ↔ ¬ ∀y ∈ a (xry ∨ yrx))
68	65, 66, 67	3bitr2i 264	. . . . . . . . . . 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))
69	11, 68	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)))
70	7, 69	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)))
71	70	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)))
72	6, 71	bitri 240	. . . . . . 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(x ∈ a ∧ ¬ ∀y ∈ a (xry ∨ yrx)))
73		df-rex 2621	. . . . . . 7 ⊢ (∃x ∈ a ¬ ∀y ∈ a (xry ∨ yrx) ↔ ∃x(x ∈ a ∧ ¬ ∀y ∈ a (xry ∨ yrx)))
74		rexnal 2626	. . . . . . 7 ⊢ (∃x ∈ a ¬ ∀y ∈ a (xry ∨ yrx) ↔ ¬ ∀x ∈ a ∀y ∈ a (xry ∨ yrx))
75	72, 73, 74	3bitr2i 264	. . . . . 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 ⊢ Connex = ∼ (( 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	unex 4107	. . . . . . . 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	difex 4108	. . . . . 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 ⊢ Connex ∈ V

Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∪ cun 3208   ∩ 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   Connex cconnex 5893

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-connex 5904

This theorem is referenced by:  strictex  5919