Theorem antisymex 5913

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

Assertion

Ref	Expression
antisymex	|- Antisym e. _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>. | A.x e. a A.y e. a ((xry /\ yrx) -> 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 Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c) <-> -. <.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c))
6		elin 3220	. . . . . . . . . 10 |- (<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c)) <-> (<.{x}, <.r, a>.>. e. Ins2 S /\ <.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"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 Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I )) <-> (<.{y}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S /\ <.{y}, <.{x}, <.r, a>.>.>. e. Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"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	3	oqelins4 5795	. . . . . . . . . . . . . . . . 17 |- (<.{y}, <.{x}, <.r, a>.>.>. e. Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ) <-> <.{y}, <.{x}, r>.>. e. (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))
19		eldif 3222	. . . . . . . . . . . . . . . . 17 |- (<.{y}, <.{x}, r>.>. e. (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ) <-> (<.{y}, <.{x}, r>.>. e. ((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) /\ -. <.{y}, <.{x}, r>.>. e. Ins3 _I ))
20		elin 3220	. . . . . . . . . . . . . . . . . . 19 |- (<.{y}, <.{x}, r>.>. e. ((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) <-> (<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) /\ <.{y}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)))
21		elin 3220	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S ) <-> (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 (2nd (x) 1st) /\ <.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S ))
22	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 (2nd (x) 1st) <-> <.{p}, <.{y}, {x}>.>. e. SI3 (2nd (x) 1st))
23		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- p e. _V
24	23, 15, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{p}, <.{y}, {x}>.>. e. SI3 (2nd (x) 1st) <-> <.p, <.y, x>.>. e. (2nd (x) 1st))
25		oteltxp 5783	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.p, <.y, x>.>. e. (2nd (x) 1st) <-> (<.p, y>. e. 2nd /\ <.p, x>. e. 1st))
26		ancom 437	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((<.p, y>. e. 2nd /\ <.p, x>. e. 1st) <-> (<.p, x>. e. 1st /\ <.p, y>. e. 2nd))
27		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (p1stx <-> <.p, x>. e. 1st)
28		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (p2ndy <-> <.p, y>. e. 2nd)
29	27, 28	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((p1stx /\ p2ndy) <-> (<.p, x>. e. 1st /\ <.p, y>. e. 2nd))
30	26, 29	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((<.p, y>. e. 2nd /\ <.p, x>. e. 1st) <-> (p1stx /\ p2ndy))
31	8, 15	op1st2nd 5791	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((p1stx /\ p2ndy) <-> p = <.x, y>.)
32	25, 30, 31	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.p, <.y, x>.>. e. (2nd (x) 1st) <-> p = <.x, y>.)
33	22, 24, 32	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 (2nd (x) 1st) <-> p = <.x, y>.)
34		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- {y} e. _V
35	34	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S <-> <.{p}, <.{x}, r>.>. e. Ins2 S )
36	12	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{p}, <.{x}, r>.>. e. Ins2 S <-> <.{p}, r>. e. S )
37	23, 2	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{p}, r>. e. S <-> p e. r)
38	35, 36, 37	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S <-> p e. r)
39	33, 38	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 (2nd (x) 1st) /\ <.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S ) <-> (p = <.x, y>. /\ p e. r))
40	21, 39	bitri 240	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S ) <-> (p = <.x, y>. /\ p e. r))
41	40	exbii 1582	. . . . . . . . . . . . . . . . . . . . 21 |- (E.p<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S ) <-> E.p(p = <.x, y>. /\ p e. r))
42		elima1c 4948	. . . . . . . . . . . . . . . . . . . . 21 |- (<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) <-> E.p<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S ))
43		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 |- (xry <-> <.x, y>. e. r)
44		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.x, y>. e. r <-> E.p(p = <.x, y>. /\ p e. r))
45	43, 44	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 |- (xry <-> E.p(p = <.x, y>. /\ p e. r))
46	41, 42, 45	3bitr4i 268	. . . . . . . . . . . . . . . . . . . 20 |- (<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) <-> xry)
47		elin 3220	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{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 ))
48	2	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 _I <-> <.{p}, <.{y}, {x}>.>. e. SI3 _I )
49	23, 15, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.{p}, <.{y}, {x}>.>. e. SI3 _I <-> <.p, <.y, x>.>. e. _I )
50		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (p _I <.y, x>. <-> <.p, <.y, x>.>. e. _I )
51	15, 8	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- <.y, x>. e. _V
52	51	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (p _I <.y, x>. <-> p = <.y, x>.)
53	49, 50, 52	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{p}, <.{y}, {x}>.>. e. SI3 _I <-> p = <.y, x>.)
54	48, 53	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 _I <-> p = <.y, x>.)
55	54, 38	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((<.{p}, <.{y}, <.{x}, r>.>.>. e. Ins4 SI3 _I /\ <.{p}, <.{y}, <.{x}, r>.>.>. e. Ins2 Ins2 S ) <-> (p = <.y, x>. /\ p e. r))
56	47, 55	bitri 240	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ) <-> (p = <.y, x>. /\ p e. r))
57	56	exbii 1582	. . . . . . . . . . . . . . . . . . . . 21 |- (E.p<.{p}, <.{y}, <.{x}, r>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 S ) <-> E.p(p = <.y, x>. /\ p e. r))
58		elima1c 4948	. . . . . . . . . . . . . . . . . . . . 21 |- (<.{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 ))
59		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 |- (yrx <-> <.y, x>. e. r)
60		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.y, x>. e. r <-> E.p(p = <.y, x>. /\ p e. r))
61	59, 60	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 |- (yrx <-> E.p(p = <.y, x>. /\ p e. r))
62	57, 58, 61	3bitr4i 268	. . . . . . . . . . . . . . . . . . . 20 |- (<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) <-> yrx)
63	46, 62	anbi12i 678	. . . . . . . . . . . . . . . . . . 19 |- ((<.{y}, <.{x}, r>.>. e. (( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) /\ <.{y}, <.{x}, r>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) <-> (xry /\ yrx))
64	20, 63	bitri 240	. . . . . . . . . . . . . . . . . 18 |- (<.{y}, <.{x}, r>.>. e. ((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) <-> (xry /\ yrx))
65	2	otelins3 5793	. . . . . . . . . . . . . . . . . . . 20 |- (<.{y}, <.{x}, r>.>. e. Ins3 _I <-> <.{y}, {x}>. e. _I )
66		df-br 4641	. . . . . . . . . . . . . . . . . . . 20 |- ({y} _I {x} <-> <.{y}, {x}>. e. _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>.>. e. Ins3 _I <-> x = y)
72	71	notbii 287	. . . . . . . . . . . . . . . . . 18 |- (-. <.{y}, <.{x}, r>.>. e. Ins3 _I <-> -. x = y)
73	64, 72	anbi12i 678	. . . . . . . . . . . . . . . . 17 |- ((<.{y}, <.{x}, r>.>. e. ((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) /\ -. <.{y}, <.{x}, r>.>. e. Ins3 _I ) <-> ((xry /\ yrx) /\ -. x = y))
74	18, 19, 73	3bitri 262	. . . . . . . . . . . . . . . 16 |- (<.{y}, <.{x}, <.r, a>.>.>. e. Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ) <-> ((xry /\ yrx) /\ -. x = y))
75	17, 74	anbi12i 678	. . . . . . . . . . . . . . 15 |- ((<.{y}, <.{x}, <.r, a>.>.>. e. Ins2 Ins2 S /\ <.{y}, <.{x}, <.r, a>.>.>. e. Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I )) <-> (y e. a /\ ((xry /\ yrx) /\ -. x = y)))
76	11, 75	bitri 240	. . . . . . . . . . . . . 14 |- (<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I )) <-> (y e. a /\ ((xry /\ yrx) /\ -. x = y)))
77	76	exbii 1582	. . . . . . . . . . . . 13 |- (E.y<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I )) <-> E.y(y e. a /\ ((xry /\ yrx) /\ -. x = y)))
78		elima1c 4948	. . . . . . . . . . . . 13 |- (<.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c) <-> E.y<.{y}, <.{x}, <.r, a>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I )))
79		df-rex 2621	. . . . . . . . . . . . 13 |- (E.y e. a ((xry /\ yrx) /\ -. x = y) <-> E.y(y e. a /\ ((xry /\ yrx) /\ -. x = y)))
80	77, 78, 79	3bitr4i 268	. . . . . . . . . . . 12 |- (<.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c) <-> E.y e. a ((xry /\ yrx) /\ -. x = y))
81		rexanali 2661	. . . . . . . . . . . 12 |- (E.y e. a ((xry /\ yrx) /\ -. x = y) <-> -. A.y e. a ((xry /\ yrx) -> x = y))
82	80, 81	bitri 240	. . . . . . . . . . 11 |- (<.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c) <-> -. A.y e. a ((xry /\ yrx) -> x = y))
83	10, 82	anbi12i 678	. . . . . . . . . 10 |- ((<.{x}, <.r, a>.>. e. Ins2 S /\ <.{x}, <.r, a>.>. e. (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c)) <-> (x e. a /\ -. A.y e. a ((xry /\ yrx) -> x = y)))
84	6, 83	bitri 240	. . . . . . . . 9 |- (<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c)) <-> (x e. a /\ -. A.y e. a ((xry /\ yrx) -> x = y)))
85	84	exbii 1582	. . . . . . . 8 |- (E.x<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c)) <-> E.x(x e. a /\ -. A.y e. a ((xry /\ yrx) -> x = y)))
86		elima1c 4948	. . . . . . . 8 |- (<.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c) <-> E.x<.{x}, <.r, a>.>. e. ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c)))
87		df-rex 2621	. . . . . . . 8 |- (E.x e. a -. A.y e. a ((xry /\ yrx) -> x = y) <-> E.x(x e. a /\ -. A.y e. a ((xry /\ yrx) -> x = y)))
88	85, 86, 87	3bitr4i 268	. . . . . . 7 |- (<.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c) <-> E.x e. a -. A.y e. a ((xry /\ yrx) -> x = y))
89		rexnal 2626	. . . . . . 7 |- (E.x e. a -. A.y e. a ((xry /\ yrx) -> x = y) <-> -. A.x e. a A.y e. a ((xry /\ yrx) -> x = y))
90	88, 89	bitri 240	. . . . . 6 |- (<.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c) <-> -. A.x e. a A.y e. a ((xry /\ yrx) -> x = y))
91	90	con2bii 322	. . . . 5 |- (A.x e. a A.y e. a ((xry /\ yrx) -> x = y) <-> -. <.r, a>. e. (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c))
92	5, 91	bitr4i 243	. . . 4 |- (<.r, a>. e. ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c) <-> A.x e. a A.y e. a ((xry /\ yrx) -> x = y))
93	92	opabbi2i 4867	. . 3 |- ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c) = {<.r, a>. | A.x e. a A.y e. a ((xry /\ yrx) -> x = y)}
94	1, 93	eqtr4i 2376	. 2 |- Antisym = ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c)
95		ssetex 4745	. . . . . 6 |- S e. _V
96	95	ins2ex 5798	. . . . 5 |- Ins2 S e. _V
97	96	ins2ex 5798	. . . . . . 7 |- Ins2 Ins2 S e. _V
98		2ndex 5113	. . . . . . . . . . . . . . 15 |- 2nd e. _V
99		1stex 4740	. . . . . . . . . . . . . . 15 |- 1st e. _V
100	98, 99	txpex 5786	. . . . . . . . . . . . . 14 |- (2nd (x) 1st) e. _V
101	100	si3ex 5807	. . . . . . . . . . . . 13 |- SI3 (2nd (x) 1st) e. _V
102	101	ins4ex 5800	. . . . . . . . . . . 12 |- Ins4 SI3 (2nd (x) 1st) e. _V
103	102, 97	inex 4106	. . . . . . . . . . 11 |- ( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S ) e. _V
104		1cex 4143	. . . . . . . . . . 11 |- 1c e. _V
105	103, 104	imaex 4748	. . . . . . . . . 10 |- (( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) e. _V
106		idex 5505	. . . . . . . . . . . . . 14 |- _I e. _V
107	106	si3ex 5807	. . . . . . . . . . . . 13 |- SI3 _I e. _V
108	107	ins4ex 5800	. . . . . . . . . . . 12 |- Ins4 SI3 _I e. _V
109	108, 97	inex 4106	. . . . . . . . . . 11 |- ( Ins4 SI3 _I i^i Ins2 Ins2 S ) e. _V
110	109, 104	imaex 4748	. . . . . . . . . 10 |- (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c) e. _V
111	105, 110	inex 4106	. . . . . . . . 9 |- ((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) e. _V
112	106	ins3ex 5799	. . . . . . . . 9 |- Ins3 _I e. _V
113	111, 112	difex 4108	. . . . . . . 8 |- (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ) e. _V
114	113	ins4ex 5800	. . . . . . 7 |- Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ) e. _V
115	97, 114	inex 4106	. . . . . 6 |- ( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I )) e. _V
116	115, 104	imaex 4748	. . . . 5 |- (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c) e. _V
117	96, 116	inex 4106	. . . 4 |- ( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c)) e. _V
118	117, 104	imaex 4748	. . 3 |- (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c) e. _V
119	118	complex 4105	. 2 |- ∼ (( Ins2 S i^i (( Ins2 Ins2 S i^i Ins4 (((( Ins4 SI3 (2nd (x) 1st) i^i Ins2 Ins2 S )"1c) i^i (( Ins4 SI3 _I i^i Ins2 Ins2 S )"1c)) \ Ins3 _I ))"1c))"1c) e. _V
120	94, 119	eqeltri 2423	1 |- Antisym 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  1stc1st 4718   S csset 4720  "cima 4723   _I cid 4764  2ndc2nd 4784   (x) 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