Theorem antisymex 5913

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

Assertion

Ref	Expression
antisymex	⊢ Antisym ∈ V

Proof of Theorem antisymex

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

Step	Hyp	Ref	Expression
1		df-antisym 5902	. . 3 ⊢ Antisym = {⟨r, a⟩ ∣ ∀x ∈ a ∀y ∈ a ((xry ∧ yrx) → x = y)}
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 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ¬ ⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c))
6		elin 3220	. . . . . . . . . 10 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) ↔ (⟨{x}, ⟨r, a⟩⟩ ∈ Ins2 S ∧ ⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 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 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) ↔ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins2 Ins2 S ∧ ⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )))
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 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I ) ↔ ⟨{y}, ⟨{x}, r⟩⟩ ∈ (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I ))
19		eldif 3222	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I ) ↔ (⟨{y}, ⟨{x}, r⟩⟩ ∈ ((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∧ ¬ ⟨{y}, ⟨{x}, r⟩⟩ ∈ Ins3 I ))
20		elin 3220	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ ((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∧ ⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)))
21		elin 3220	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) ↔ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 (2nd ⊗ 1st ) ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ))
22	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 (2nd ⊗ 1st ) ↔ ⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 (2nd ⊗ 1st ))
23		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ p ∈ V
24	23, 15, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 (2nd ⊗ 1st ) ↔ ⟨p, ⟨y, x⟩⟩ ∈ (2nd ⊗ 1st ))
25		oteltxp 5783	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨p, ⟨y, x⟩⟩ ∈ (2nd ⊗ 1st ) ↔ (⟨p, y⟩ ∈ 2nd ∧ ⟨p, x⟩ ∈ 1st ))
26		ancom 437	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((⟨p, y⟩ ∈ 2nd ∧ ⟨p, x⟩ ∈ 1st ) ↔ (⟨p, x⟩ ∈ 1st ∧ ⟨p, y⟩ ∈ 2nd ))
27		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (p1st x ↔ ⟨p, x⟩ ∈ 1st )
28		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (p2nd y ↔ ⟨p, y⟩ ∈ 2nd )
29	27, 28	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((p1st x ∧ p2nd y) ↔ (⟨p, x⟩ ∈ 1st ∧ ⟨p, y⟩ ∈ 2nd ))
30	26, 29	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((⟨p, y⟩ ∈ 2nd ∧ ⟨p, x⟩ ∈ 1st ) ↔ (p1st x ∧ p2nd y))
31	8, 15	op1st2nd 5791	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((p1st x ∧ p2nd y) ↔ p = ⟨x, y⟩)
32	25, 30, 31	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨p, ⟨y, x⟩⟩ ∈ (2nd ⊗ 1st ) ↔ p = ⟨x, y⟩)
33	22, 24, 32	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 (2nd ⊗ 1st ) ↔ p = ⟨x, y⟩)
34		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {y} ∈ V
35	34	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{p}, ⟨{x}, r⟩⟩ ∈ Ins2 S )
36	12	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{p}, ⟨{x}, r⟩⟩ ∈ Ins2 S ↔ ⟨{p}, r⟩ ∈ S )
37	23, 2	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{p}, r⟩ ∈ S ↔ p ∈ r)
38	35, 36, 37	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ↔ p ∈ r)
39	33, 38	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 (2nd ⊗ 1st ) ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ) ↔ (p = ⟨x, y⟩ ∧ p ∈ r))
40	21, 39	bitri 240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) ↔ (p = ⟨x, y⟩ ∧ p ∈ r))
41	40	exbii 1582	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) ↔ ∃p(p = ⟨x, y⟩ ∧ p ∈ r))
42		elima1c 4948	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ↔ ∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ))
43		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (xry ↔ ⟨x, y⟩ ∈ r)
44		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨x, y⟩ ∈ r ↔ ∃p(p = ⟨x, y⟩ ∧ p ∈ r))
45	43, 44	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (xry ↔ ∃p(p = ⟨x, y⟩ ∧ p ∈ r))
46	41, 42, 45	3bitr4i 268	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ↔ xry)
47		elin 3220	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ))
48	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ↔ ⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 I )
49	23, 15, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 I ↔ ⟨p, ⟨y, x⟩⟩ ∈ I )
50		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (p I ⟨y, x⟩ ↔ ⟨p, ⟨y, x⟩⟩ ∈ I )
51	15, 8	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ⟨y, x⟩ ∈ V
52	51	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (p I ⟨y, x⟩ ↔ p = ⟨y, x⟩)
53	49, 50, 52	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{p}, ⟨{y}, {x}⟩⟩ ∈ SI3 I ↔ p = ⟨y, x⟩)
54	48, 53	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ↔ p = ⟨y, x⟩)
55	54, 38	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins4 SI3 I ∧ ⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ Ins2 Ins2 S ) ↔ (p = ⟨y, x⟩ ∧ p ∈ r))
56	47, 55	bitri 240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ (p = ⟨y, x⟩ ∧ p ∈ r))
57	56	exbii 1582	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ ∃p(p = ⟨y, x⟩ ∧ p ∈ r))
58		elima1c 4948	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ ∃p⟨{p}, ⟨{y}, ⟨{x}, r⟩⟩⟩ ∈ ( Ins4 SI3 I ∩ Ins2 Ins2 S ))
59		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (yrx ↔ ⟨y, x⟩ ∈ r)
60		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨y, x⟩ ∈ r ↔ ∃p(p = ⟨y, x⟩ ∧ p ∈ r))
61	59, 60	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (yrx ↔ ∃p(p = ⟨y, x⟩ ∧ p ∈ r))
62	57, 58, 61	3bitr4i 268	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ yrx)
63	46, 62	anbi12i 678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∧ ⟨{y}, ⟨{x}, r⟩⟩ ∈ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (xry ∧ yrx))
64	20, 63	bitri 240	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ ((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (xry ∧ yrx))
65	2	otelins3 5793	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ Ins3 I ↔ ⟨{y}, {x}⟩ ∈ I )
66		df-br 4641	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({y} I {x} ↔ ⟨{y}, {x}⟩ ∈ I )
67	12	ideq 4871	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({y} I {x} ↔ {y} = {x})
68		eqcom 2355	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({y} = {x} ↔ {x} = {y})
69	8	sneqb 3877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({x} = {y} ↔ x = y)
70	67, 68, 69	3bitri 262	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({y} I {x} ↔ x = y)
71	65, 66, 70	3bitr2i 264	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{y}, ⟨{x}, r⟩⟩ ∈ Ins3 I ↔ x = y)
72	71	notbii 287	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ⟨{y}, ⟨{x}, r⟩⟩ ∈ Ins3 I ↔ ¬ x = y)
73	64, 72	anbi12i 678	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨{y}, ⟨{x}, r⟩⟩ ∈ ((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∧ ¬ ⟨{y}, ⟨{x}, r⟩⟩ ∈ Ins3 I ) ↔ ((xry ∧ yrx) ∧ ¬ x = y))
74	18, 19, 73	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I ) ↔ ((xry ∧ yrx) ∧ ¬ x = y))
75	17, 74	anbi12i 678	. . . . . . . . . . . . . . 15 ⊢ ((⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins2 Ins2 S ∧ ⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) ↔ (y ∈ a ∧ ((xry ∧ yrx) ∧ ¬ x = y)))
76	11, 75	bitri 240	. . . . . . . . . . . . . 14 ⊢ (⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) ↔ (y ∈ a ∧ ((xry ∧ yrx) ∧ ¬ x = y)))
77	76	exbii 1582	. . . . . . . . . . . . 13 ⊢ (∃y⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) ↔ ∃y(y ∈ a ∧ ((xry ∧ yrx) ∧ ¬ x = y)))
78		elima1c 4948	. . . . . . . . . . . . 13 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c) ↔ ∃y⟨{y}, ⟨{x}, ⟨r, a⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )))
79		df-rex 2621	. . . . . . . . . . . . 13 ⊢ (∃y ∈ a ((xry ∧ yrx) ∧ ¬ x = y) ↔ ∃y(y ∈ a ∧ ((xry ∧ yrx) ∧ ¬ x = y)))
80	77, 78, 79	3bitr4i 268	. . . . . . . . . . . 12 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c) ↔ ∃y ∈ a ((xry ∧ yrx) ∧ ¬ x = y))
81		rexanali 2661	. . . . . . . . . . . 12 ⊢ (∃y ∈ a ((xry ∧ yrx) ∧ ¬ x = y) ↔ ¬ ∀y ∈ a ((xry ∧ yrx) → x = y))
82	80, 81	bitri 240	. . . . . . . . . . 11 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c) ↔ ¬ ∀y ∈ a ((xry ∧ yrx) → x = y))
83	10, 82	anbi12i 678	. . . . . . . . . 10 ⊢ ((⟨{x}, ⟨r, a⟩⟩ ∈ Ins2 S ∧ ⟨{x}, ⟨r, a⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) ↔ (x ∈ a ∧ ¬ ∀y ∈ a ((xry ∧ yrx) → x = y)))
84	6, 83	bitri 240	. . . . . . . . 9 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) ↔ (x ∈ a ∧ ¬ ∀y ∈ a ((xry ∧ yrx) → x = y)))
85	84	exbii 1582	. . . . . . . 8 ⊢ (∃x⟨{x}, ⟨r, a⟩⟩ ∈ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) ↔ ∃x(x ∈ a ∧ ¬ ∀y ∈ a ((xry ∧ yrx) → x = y)))
86		elima1c 4948	. . . . . . . 8 ⊢ (⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ∃x⟨{x}, ⟨r, a⟩⟩ ∈ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)))
87		df-rex 2621	. . . . . . . 8 ⊢ (∃x ∈ a ¬ ∀y ∈ a ((xry ∧ yrx) → x = y) ↔ ∃x(x ∈ a ∧ ¬ ∀y ∈ a ((xry ∧ yrx) → x = y)))
88	85, 86, 87	3bitr4i 268	. . . . . . 7 ⊢ (⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ∃x ∈ a ¬ ∀y ∈ a ((xry ∧ yrx) → x = y))
89		rexnal 2626	. . . . . . 7 ⊢ (∃x ∈ a ¬ ∀y ∈ a ((xry ∧ yrx) → x = y) ↔ ¬ ∀x ∈ a ∀y ∈ a ((xry ∧ yrx) → x = y))
90	88, 89	bitri 240	. . . . . 6 ⊢ (⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ¬ ∀x ∈ a ∀y ∈ a ((xry ∧ yrx) → x = y))
91	90	con2bii 322	. . . . 5 ⊢ (∀x ∈ a ∀y ∈ a ((xry ∧ yrx) → x = y) ↔ ¬ ⟨r, a⟩ ∈ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c))
92	5, 91	bitr4i 243	. . . 4 ⊢ (⟨r, a⟩ ∈ ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c) ↔ ∀x ∈ a ∀y ∈ a ((xry ∧ yrx) → x = y))
93	92	opabbi2i 4867	. . 3 ⊢ ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c) = {⟨r, a⟩ ∣ ∀x ∈ a ∀y ∈ a ((xry ∧ yrx) → x = y)}
94	1, 93	eqtr4i 2376	. 2 ⊢ Antisym = ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c)
95		ssetex 4745	. . . . . 6 ⊢ S ∈ V
96	95	ins2ex 5798	. . . . 5 ⊢ Ins2 S ∈ V
97	96	ins2ex 5798	. . . . . . 7 ⊢ Ins2 Ins2 S ∈ V
98		2ndex 5113	. . . . . . . . . . . . . . 15 ⊢ 2nd ∈ V
99		1stex 4740	. . . . . . . . . . . . . . 15 ⊢ 1st ∈ V
100	98, 99	txpex 5786	. . . . . . . . . . . . . 14 ⊢ (2nd ⊗ 1st ) ∈ V
101	100	si3ex 5807	. . . . . . . . . . . . 13 ⊢ SI3 (2nd ⊗ 1st ) ∈ V
102	101	ins4ex 5800	. . . . . . . . . . . 12 ⊢ Ins4 SI3 (2nd ⊗ 1st ) ∈ V
103	102, 97	inex 4106	. . . . . . . . . . 11 ⊢ ( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) ∈ V
104		1cex 4143	. . . . . . . . . . 11 ⊢ 1c ∈ V
105	103, 104	imaex 4748	. . . . . . . . . 10 ⊢ (( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∈ V
106		idex 5505	. . . . . . . . . . . . . 14 ⊢ I ∈ V
107	106	si3ex 5807	. . . . . . . . . . . . 13 ⊢ SI3 I ∈ V
108	107	ins4ex 5800	. . . . . . . . . . . 12 ⊢ Ins4 SI3 I ∈ V
109	108, 97	inex 4106	. . . . . . . . . . 11 ⊢ ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ∈ V
110	109, 104	imaex 4748	. . . . . . . . . 10 ⊢ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ∈ V
111	105, 110	inex 4106	. . . . . . . . 9 ⊢ ((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∈ V
112	106	ins3ex 5799	. . . . . . . . 9 ⊢ Ins3 I ∈ V
113	111, 112	difex 4108	. . . . . . . 8 ⊢ (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I ) ∈ V
114	113	ins4ex 5800	. . . . . . 7 ⊢ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I ) ∈ V
115	97, 114	inex 4106	. . . . . 6 ⊢ ( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) ∈ V
116	115, 104	imaex 4748	. . . . 5 ⊢ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c) ∈ V
117	96, 116	inex 4106	. . . 4 ⊢ ( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) ∈ V
118	117, 104	imaex 4748	. . 3 ⊢ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c) ∈ V
119	118	complex 4105	. 2 ⊢ ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ∖ Ins3 I )) “ 1c)) “ 1c) ∈ V
120	94, 119	eqeltri 2423	1 ⊢ Antisym ∈ 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  1^st c1st 4718   S csset 4720   “ cima 4723   I cid 4764  2^nd c2nd 4784   ⊗ ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752   Ins4 cins4 5756   SI₃ csi3 5758   Antisym cantisym 5891

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-antisym 5902

This theorem is referenced by:  partialex  5918