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Theorem antisymex 5912
 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.
StepHypRef Expression
1 df-antisym 5901 . . 3 Antisym = {r, a x a y a ((xry yrx) → x = y)}
2 vex 2862 . . . . . . 7 r V
3 vex 2862 . . . . . . 7 a V
42, 3opex 4588 . . . . . 6 r, a V
54elcompl 3225 . . . . 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 3219 . . . . . . . . . 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)))
72otelins2 5791 . . . . . . . . . . . 12 ({x}, r, a Ins2 S {x}, a S )
8 vex 2862 . . . . . . . . . . . . 13 x V
98, 3opelssetsn 4760 . . . . . . . . . . . 12 ({x}, a S x a)
107, 9bitri 240 . . . . . . . . . . 11 ({x}, r, a Ins2 S x a)
11 elin 3219 . . . . . . . . . . . . . . 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 4111 . . . . . . . . . . . . . . . . . 18 {x} V
1312otelins2 5791 . . . . . . . . . . . . . . . . 17 ({y}, {x}, r, a Ins2 Ins2 S {y}, r, a Ins2 S )
142otelins2 5791 . . . . . . . . . . . . . . . . 17 ({y}, r, a Ins2 S {y}, a S )
15 vex 2862 . . . . . . . . . . . . . . . . . 18 y V
1615, 3opelssetsn 4760 . . . . . . . . . . . . . . . . 17 ({y}, a S y a)
1713, 14, 163bitri 262 . . . . . . . . . . . . . . . 16 ({y}, {x}, r, a Ins2 Ins2 S y a)
183oqelins4 5794 . . . . . . . . . . . . . . . . 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 3221 . . . . . . . . . . . . . . . . 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 3219 . . . . . . . . . . . . . . . . . . 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 3219 . . . . . . . . . . . . . . . . . . . . . . 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 ))
222oqelins4 5794 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({p}, {y}, {x}, r Ins4 SI3 (2nd ⊗ 1st ) ↔ {p}, {y}, {x} SI3 (2nd ⊗ 1st ))
23 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . 26 p V
2423, 15, 8otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({p}, {y}, {x} SI3 (2nd ⊗ 1st ) ↔ p, y, x (2nd ⊗ 1st ))
25 oteltxp 5782 . . . . . . . . . . . . . . . . . . . . . . . . . 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 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (p1st xp, x 1st )
28 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (p2nd yp, y 2nd )
2927, 28anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((p1st x p2nd y) ↔ (p, x 1st p, y 2nd ))
3026, 29bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((p, y 2nd p, x 1st ) ↔ (p1st x p2nd y))
318, 15op1st2nd 5790 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((p1st x p2nd y) ↔ p = x, y)
3225, 30, 313bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . 25 (p, y, x (2nd ⊗ 1st ) ↔ p = x, y)
3322, 24, 323bitri 262 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, {y}, {x}, r Ins4 SI3 (2nd ⊗ 1st ) ↔ p = x, y)
34 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {y} V
3534otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({p}, {y}, {x}, r Ins2 Ins2 S {p}, {x}, r Ins2 S )
3612otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({p}, {x}, r Ins2 S {p}, r S )
3723, 2opelssetsn 4760 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({p}, r S p r)
3835, 36, 373bitri 262 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, {y}, {x}, r Ins2 Ins2 S p r)
3933, 38anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . 23 (({p}, {y}, {x}, r Ins4 SI3 (2nd ⊗ 1st ) {p}, {y}, {x}, r Ins2 Ins2 S ) ↔ (p = x, y p r))
4021, 39bitri 240 . . . . . . . . . . . . . . . . . . . . . 22 ({p}, {y}, {x}, r ( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) ↔ (p = x, y p r))
4140exbii 1582 . . . . . . . . . . . . . . . . . . . . 21 (p{p}, {y}, {x}, r ( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) ↔ p(p = x, y p r))
42 elima1c 4947 . . . . . . . . . . . . . . . . . . . . 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 4640 . . . . . . . . . . . . . . . . . . . . . 22 (xryx, y r)
44 df-clel 2349 . . . . . . . . . . . . . . . . . . . . . 22 (x, y rp(p = x, y p r))
4543, 44bitri 240 . . . . . . . . . . . . . . . . . . . . 21 (xryp(p = x, y p r))
4641, 42, 453bitr4i 268 . . . . . . . . . . . . . . . . . . . 20 ({y}, {x}, r (( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ↔ xry)
47 elin 3219 . . . . . . . . . . . . . . . . . . . . . . 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 ))
482oqelins4 5794 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({p}, {y}, {x}, r Ins4 SI3 I ↔ {p}, {y}, {x} SI3 I )
4923, 15, 8otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({p}, {y}, {x} SI3 I ↔ p, y, x I )
50 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (p I y, xp, y, x I )
5115, 8opex 4588 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 y, x V
5251ideq 4870 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (p I y, xp = y, x)
5349, 50, 523bitr2i 264 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({p}, {y}, {x} SI3 I ↔ p = y, x)
5448, 53bitri 240 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, {y}, {x}, r Ins4 SI3 I ↔ p = y, x)
5554, 38anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . 23 (({p}, {y}, {x}, r Ins4 SI3 I {p}, {y}, {x}, r Ins2 Ins2 S ) ↔ (p = y, x p r))
5647, 55bitri 240 . . . . . . . . . . . . . . . . . . . . . 22 ({p}, {y}, {x}, r ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ (p = y, x p r))
5756exbii 1582 . . . . . . . . . . . . . . . . . . . . 21 (p{p}, {y}, {x}, r ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ p(p = y, x p r))
58 elima1c 4947 . . . . . . . . . . . . . . . . . . . . 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 4640 . . . . . . . . . . . . . . . . . . . . . 22 (yrxy, x r)
60 df-clel 2349 . . . . . . . . . . . . . . . . . . . . . 22 (y, x rp(p = y, x p r))
6159, 60bitri 240 . . . . . . . . . . . . . . . . . . . . 21 (yrxp(p = y, x p r))
6257, 58, 613bitr4i 268 . . . . . . . . . . . . . . . . . . . 20 ({y}, {x}, r (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ yrx)
6346, 62anbi12i 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))
6420, 63bitri 240 . . . . . . . . . . . . . . . . . 18 ({y}, {x}, r ((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (xry yrx))
652otelins3 5792 . . . . . . . . . . . . . . . . . . . 20 ({y}, {x}, r Ins3 I ↔ {y}, {x} I )
66 df-br 4640 . . . . . . . . . . . . . . . . . . . 20 ({y} I {x} ↔ {y}, {x} I )
6712ideq 4870 . . . . . . . . . . . . . . . . . . . . 21 ({y} I {x} ↔ {y} = {x})
68 eqcom 2355 . . . . . . . . . . . . . . . . . . . . 21 ({y} = {x} ↔ {x} = {y})
698sneqb 3876 . . . . . . . . . . . . . . . . . . . . 21 ({x} = {y} ↔ x = y)
7067, 68, 693bitri 262 . . . . . . . . . . . . . . . . . . . 20 ({y} I {x} ↔ x = y)
7165, 66, 703bitr2i 264 . . . . . . . . . . . . . . . . . . 19 ({y}, {x}, r Ins3 I ↔ x = y)
7271notbii 287 . . . . . . . . . . . . . . . . . 18 {y}, {x}, r Ins3 I ↔ ¬ x = y)
7364, 72anbi12i 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))
7418, 19, 733bitri 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))
7517, 74anbi12i 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)))
7611, 75bitri 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)))
7776exbii 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 4947 . . . . . . . . . . . . 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 2620 . . . . . . . . . . . . 13 (y a ((xry yrx) ¬ x = y) ↔ y(y a ((xry yrx) ¬ x = y)))
8077, 78, 793bitr4i 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 2660 . . . . . . . . . . . 12 (y a ((xry yrx) ¬ x = y) ↔ ¬ y a ((xry yrx) → x = y))
8280, 81bitri 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))
8310, 82anbi12i 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)))
846, 83bitri 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)))
8584exbii 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 4947 . . . . . . . 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 2620 . . . . . . . 8 (x a ¬ y a ((xry yrx) → x = y) ↔ x(x a ¬ y a ((xry yrx) → x = y)))
8885, 86, 873bitr4i 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 2625 . . . . . . 7 (x a ¬ y a ((xry yrx) → x = y) ↔ ¬ x a y a ((xry yrx) → x = y))
9088, 89bitri 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))
9190con2bii 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))
925, 91bitr4i 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))
9392opabbi2i 4866 . . 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)}
941, 93eqtr4i 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 4744 . . . . . 6 S V
9695ins2ex 5797 . . . . 5 Ins2 S V
9796ins2ex 5797 . . . . . . 7 Ins2 Ins2 S V
98 2ndex 5112 . . . . . . . . . . . . . . 15 2nd V
99 1stex 4739 . . . . . . . . . . . . . . 15 1st V
10098, 99txpex 5785 . . . . . . . . . . . . . 14 (2nd ⊗ 1st ) V
101100si3ex 5806 . . . . . . . . . . . . 13 SI3 (2nd ⊗ 1st ) V
102101ins4ex 5799 . . . . . . . . . . . 12 Ins4 SI3 (2nd ⊗ 1st ) V
103102, 97inex 4105 . . . . . . . . . . 11 ( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) V
104 1cex 4142 . . . . . . . . . . 11 1c V
105103, 104imaex 4747 . . . . . . . . . 10 (( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) V
106 idex 5504 . . . . . . . . . . . . . 14 I V
107106si3ex 5806 . . . . . . . . . . . . 13 SI3 I V
108107ins4ex 5799 . . . . . . . . . . . 12 Ins4 SI3 I V
109108, 97inex 4105 . . . . . . . . . . 11 ( Ins4 SI3 I ∩ Ins2 Ins2 S ) V
110109, 104imaex 4747 . . . . . . . . . 10 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) V
111105, 110inex 4105 . . . . . . . . 9 ((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) V
112106ins3ex 5798 . . . . . . . . 9 Ins3 I V
113111, 112difex 4107 . . . . . . . 8 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) Ins3 I ) V
114113ins4ex 5799 . . . . . . 7 Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) Ins3 I ) V
11597, 114inex 4105 . . . . . 6 ( Ins2 Ins2 S Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) Ins3 I )) V
116115, 104imaex 4747 . . . . 5 (( Ins2 Ins2 S Ins4 (((( Ins4 SI3 (2nd ⊗ 1st ) ∩ Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) Ins3 I )) “ 1c) V
11796, 116inex 4105 . . . 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
118117, 104imaex 4747 . . 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
119118complex 4104 . 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
12094, 119eqeltri 2423 1 Antisym V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206   ∩ cin 3208  {csn 3737  1cc1c 4134  ⟨cop 4561  {copab 4622   class class class wbr 4639  1st c1st 4717   S csset 4719   “ cima 4722   I cid 4763  2nd c2nd 4783   ⊗ ctxp 5735   Ins2 cins2 5749   Ins3 cins3 5751   Ins4 cins4 5755   SI3 csi3 5757   Antisym cantisym 5890 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-cnv 4785  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758  df-antisym 5901 This theorem is referenced by:  partialex  5917
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