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Theorem extex 5915
 Description: The class of all extensional relationships is a set. (Contributed by SF, 19-Feb-2015.)
Assertion
Ref Expression
extex Ext V

Proof of Theorem extex
Dummy variables p a r x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ext 5907 . . 3 Ext = {r, a x a y a (z a (zrxzry) → 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 ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c) ↔ ¬ r, a (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c))
6 elin 3219 . . . . . . . . . 10 ({x}, r, a ( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) ↔ ({x}, r, a Ins2 S {x}, r, a (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 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 ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) ↔ ({y}, {x}, r, a Ins2 Ins2 S {y}, {x}, r, a ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 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)
18 eldif 3221 . . . . . . . . . . . . . . . . 17 ({y}, {x}, r, a ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I ) ↔ ({y}, {x}, r, a ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) ¬ {y}, {x}, r, a Ins3 I ))
19 snex 4111 . . . . . . . . . . . . . . . . . . . . . 22 {y} V
2012, 4opex 4588 . . . . . . . . . . . . . . . . . . . . . 22 {x}, r, a V
2119, 20opex 4588 . . . . . . . . . . . . . . . . . . . . 21 {y}, {x}, r, a V
2221elcompl 3225 . . . . . . . . . . . . . . . . . . . 20 ({y}, {x}, r, a ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) ↔ ¬ {y}, {x}, r, a (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c))
23 elin 3219 . . . . . . . . . . . . . . . . . . . . . . 23 ({z}, {y}, {x}, r, a ( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) ↔ ({z}, {y}, {x}, r, a Ins2 Ins2 Ins2 S {z}, {y}, {x}, r, a ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))))
2419otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({z}, {y}, {x}, r, a Ins2 Ins2 Ins2 S {z}, {x}, r, a Ins2 Ins2 S )
2512otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({z}, {x}, r, a Ins2 Ins2 S {z}, r, a Ins2 S )
262otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({z}, r, a Ins2 S {z}, a S )
27 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 z V
2827, 3opelssetsn 4760 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({z}, a S z a)
2926, 28bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({z}, r, a Ins2 S z a)
3024, 25, 293bitri 262 . . . . . . . . . . . . . . . . . . . . . . . 24 ({z}, {y}, {x}, r, a Ins2 Ins2 Ins2 S z a)
31 elsymdif 3223 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({z}, {y}, {x}, r, a ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) ↔ ¬ ({z}, {y}, {x}, r, a Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ {z}, {y}, {x}, r, a (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c)))
3219otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({z}, {y}, {x}, r, a Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ {z}, {x}, r, a Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))
33 elima1c 4947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ({z}, {x}, r (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ p{p}, {z}, {x}, r ( Ins4 SI3 I ∩ Ins2 Ins2 S ))
34 elin 3219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ({p}, {z}, {x}, r ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ ({p}, {z}, {x}, r Ins4 SI3 I {p}, {z}, {x}, r Ins2 Ins2 S ))
352oqelins4 5794 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ({p}, {z}, {x}, r Ins4 SI3 I ↔ {p}, {z}, {x} SI3 I )
36 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 p V
3736, 27, 8otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ({p}, {z}, {x} SI3 I ↔ p, z, x I )
38 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (p I z, xp, z, x I )
3927, 8opex 4588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 z, x V
4039ideq 4870 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (p I z, xp = z, x)
4137, 38, 403bitr2i 264 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ({p}, {z}, {x} SI3 I ↔ p = z, x)
4235, 41bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ({p}, {z}, {x}, r Ins4 SI3 I ↔ p = z, x)
43 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 {z} V
4443otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ({p}, {z}, {x}, r Ins2 Ins2 S {p}, {x}, r Ins2 S )
4512otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ({p}, {x}, r Ins2 S {p}, r S )
4636, 2opelssetsn 4760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ({p}, r S p r)
4744, 45, 463bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ({p}, {z}, {x}, r Ins2 Ins2 S p r)
4842, 47anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (({p}, {z}, {x}, r Ins4 SI3 I {p}, {z}, {x}, r Ins2 Ins2 S ) ↔ (p = z, x p r))
4934, 48bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ({p}, {z}, {x}, r ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ (p = z, x p r))
5049exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (p{p}, {z}, {x}, r ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ p(p = z, x p r))
5133, 50bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ({z}, {x}, r (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ p(p = z, x p r))
523oqelins4 5794 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ({z}, {x}, r, a Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ {z}, {x}, r (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))
53 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (zrxz, x r)
54 df-clel 2349 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (z, x rp(p = z, x p r))
5553, 54bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (zrxp(p = z, x p r))
5651, 52, 553bitr4i 268 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({z}, {x}, r, a Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ zrx)
5732, 56bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({z}, {y}, {x}, r, a Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ zrx)
58 elin 3219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ({p}, {z}, {y}, {x}, r, a ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) ↔ ({p}, {z}, {y}, {x}, r, a Ins4 SI3 I {p}, {z}, {y}, {x}, r, a Ins2 Ins2 Ins2 Ins3 S ))
5920oqelins4 5794 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ({p}, {z}, {y}, {x}, r, a Ins4 SI3 I ↔ {p}, {z}, {y} SI3 I )
6036, 27, 15otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ({p}, {z}, {y} SI3 I ↔ p, z, y I )
61 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (p I z, yp, z, y I )
6227, 15opex 4588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 z, y V
6362ideq 4870 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (p I z, yp = z, y)
6460, 61, 633bitr2i 264 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ({p}, {z}, {y} SI3 I ↔ p = z, y)
6559, 64bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ({p}, {z}, {y}, {x}, r, a Ins4 SI3 I ↔ p = z, y)
6643otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ({p}, {z}, {y}, {x}, r, a Ins2 Ins2 Ins2 Ins3 S {p}, {y}, {x}, r, a Ins2 Ins2 Ins3 S )
6719otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ({p}, {y}, {x}, r, a Ins2 Ins2 Ins3 S {p}, {x}, r, a Ins2 Ins3 S )
6812otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ({p}, {x}, r, a Ins2 Ins3 S {p}, r, a Ins3 S )
693otelins3 5792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ({p}, r, a Ins3 S {p}, r S )
7068, 69, 463bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ({p}, {x}, r, a Ins2 Ins3 S p r)
7166, 67, 703bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ({p}, {z}, {y}, {x}, r, a Ins2 Ins2 Ins2 Ins3 S p r)
7265, 71anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (({p}, {z}, {y}, {x}, r, a Ins4 SI3 I {p}, {z}, {y}, {x}, r, a Ins2 Ins2 Ins2 Ins3 S ) ↔ (p = z, y p r))
7358, 72bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ({p}, {z}, {y}, {x}, r, a ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) ↔ (p = z, y p r))
7473exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (p{p}, {z}, {y}, {x}, r, a ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) ↔ p(p = z, y p r))
75 elima1c 4947 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({z}, {y}, {x}, r, a (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c) ↔ p{p}, {z}, {y}, {x}, r, a ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ))
76 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (zryz, y r)
77 df-clel 2349 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (z, y rp(p = z, y p r))
7876, 77bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (zryp(p = z, y p r))
7974, 75, 783bitr4i 268 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({z}, {y}, {x}, r, a (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c) ↔ zry)
8057, 79bibi12i 306 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({z}, {y}, {x}, r, a Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ {z}, {y}, {x}, r, a (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) ↔ (zrxzry))
8131, 80xchbinx 301 . . . . . . . . . . . . . . . . . . . . . . . 24 ({z}, {y}, {x}, r, a ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) ↔ ¬ (zrxzry))
8230, 81anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . 23 (({z}, {y}, {x}, r, a Ins2 Ins2 Ins2 S {z}, {y}, {x}, r, a ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) ↔ (z a ¬ (zrxzry)))
8323, 82bitri 240 . . . . . . . . . . . . . . . . . . . . . 22 ({z}, {y}, {x}, r, a ( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) ↔ (z a ¬ (zrxzry)))
8483exbii 1582 . . . . . . . . . . . . . . . . . . . . 21 (z{z}, {y}, {x}, r, a ( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) ↔ z(z a ¬ (zrxzry)))
85 elima1c 4947 . . . . . . . . . . . . . . . . . . . . 21 ({y}, {x}, r, a (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) ↔ z{z}, {y}, {x}, r, a ( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))))
86 df-rex 2620 . . . . . . . . . . . . . . . . . . . . 21 (z a ¬ (zrxzry) ↔ z(z a ¬ (zrxzry)))
8784, 85, 863bitr4i 268 . . . . . . . . . . . . . . . . . . . 20 ({y}, {x}, r, a (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) ↔ z a ¬ (zrxzry))
8822, 87xchbinx 301 . . . . . . . . . . . . . . . . . . 19 ({y}, {x}, r, a ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) ↔ ¬ z a ¬ (zrxzry))
89 dfral2 2626 . . . . . . . . . . . . . . . . . . 19 (z a (zrxzry) ↔ ¬ z a ¬ (zrxzry))
9088, 89bitr4i 243 . . . . . . . . . . . . . . . . . 18 ({y}, {x}, r, a ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) ↔ z a (zrxzry))
914otelins3 5792 . . . . . . . . . . . . . . . . . . . 20 ({y}, {x}, r, a Ins3 I ↔ {y}, {x} I )
92 df-br 4640 . . . . . . . . . . . . . . . . . . . 20 ({y} I {x} ↔ {y}, {x} I )
9312ideq 4870 . . . . . . . . . . . . . . . . . . . . 21 ({y} I {x} ↔ {y} = {x})
9415sneqb 3876 . . . . . . . . . . . . . . . . . . . . 21 ({y} = {x} ↔ y = x)
95 equcom 1680 . . . . . . . . . . . . . . . . . . . . 21 (y = xx = y)
9693, 94, 953bitri 262 . . . . . . . . . . . . . . . . . . . 20 ({y} I {x} ↔ x = y)
9791, 92, 963bitr2i 264 . . . . . . . . . . . . . . . . . . 19 ({y}, {x}, r, a Ins3 I ↔ x = y)
9897notbii 287 . . . . . . . . . . . . . . . . . 18 {y}, {x}, r, a Ins3 I ↔ ¬ x = y)
9990, 98anbi12i 678 . . . . . . . . . . . . . . . . 17 (({y}, {x}, r, a ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) ¬ {y}, {x}, r, a Ins3 I ) ↔ (z a (zrxzry) ¬ x = y))
10018, 99bitri 240 . . . . . . . . . . . . . . . 16 ({y}, {x}, r, a ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I ) ↔ (z a (zrxzry) ¬ x = y))
10117, 100anbi12i 678 . . . . . . . . . . . . . . 15 (({y}, {x}, r, a Ins2 Ins2 S {y}, {x}, r, a ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) ↔ (y a (z a (zrxzry) ¬ x = y)))
10211, 101bitri 240 . . . . . . . . . . . . . 14 ({y}, {x}, r, a ( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) ↔ (y a (z a (zrxzry) ¬ x = y)))
103102exbii 1582 . . . . . . . . . . . . 13 (y{y}, {x}, r, a ( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) ↔ y(y a (z a (zrxzry) ¬ x = y)))
104 elima1c 4947 . . . . . . . . . . . . 13 ({x}, r, a (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c) ↔ y{y}, {x}, r, a ( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )))
105 df-rex 2620 . . . . . . . . . . . . 13 (y a (z a (zrxzry) ¬ x = y) ↔ y(y a (z a (zrxzry) ¬ x = y)))
106103, 104, 1053bitr4i 268 . . . . . . . . . . . 12 ({x}, r, a (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c) ↔ y a (z a (zrxzry) ¬ x = y))
107 rexanali 2660 . . . . . . . . . . . 12 (y a (z a (zrxzry) ¬ x = y) ↔ ¬ y a (z a (zrxzry) → x = y))
108106, 107bitri 240 . . . . . . . . . . 11 ({x}, r, a (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c) ↔ ¬ y a (z a (zrxzry) → x = y))
10910, 108anbi12i 678 . . . . . . . . . 10 (({x}, r, a Ins2 S {x}, r, a (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) ↔ (x a ¬ y a (z a (zrxzry) → x = y)))
1106, 109bitri 240 . . . . . . . . 9 ({x}, r, a ( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) ↔ (x a ¬ y a (z a (zrxzry) → x = y)))
111110exbii 1582 . . . . . . . 8 (x{x}, r, a ( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) ↔ x(x a ¬ y a (z a (zrxzry) → x = y)))
112 elima1c 4947 . . . . . . . 8 (r, a (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c) ↔ x{x}, r, a ( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)))
113 df-rex 2620 . . . . . . . 8 (x a ¬ y a (z a (zrxzry) → x = y) ↔ x(x a ¬ y a (z a (zrxzry) → x = y)))
114111, 112, 1133bitr4i 268 . . . . . . 7 (r, a (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c) ↔ x a ¬ y a (z a (zrxzry) → x = y))
115 rexnal 2625 . . . . . . 7 (x a ¬ y a (z a (zrxzry) → x = y) ↔ ¬ x a y a (z a (zrxzry) → x = y))
116114, 115bitri 240 . . . . . 6 (r, a (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c) ↔ ¬ x a y a (z a (zrxzry) → x = y))
117116con2bii 322 . . . . 5 (x a y a (z a (zrxzry) → x = y) ↔ ¬ r, a (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c))
1185, 117bitr4i 243 . . . 4 (r, a ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c) ↔ x a y a (z a (zrxzry) → x = y))
119118opabbi2i 4866 . . 3 ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c) = {r, a x a y a (z a (zrxzry) → x = y)}
1201, 119eqtr4i 2376 . 2 Ext = ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c)
121 ssetex 4744 . . . . . 6 S V
122121ins2ex 5797 . . . . 5 Ins2 S V
123122ins2ex 5797 . . . . . . 7 Ins2 Ins2 S V
124123ins2ex 5797 . . . . . . . . . . 11 Ins2 Ins2 Ins2 S V
125 idex 5504 . . . . . . . . . . . . . . . . . 18 I V
126125si3ex 5806 . . . . . . . . . . . . . . . . 17 SI3 I V
127126ins4ex 5799 . . . . . . . . . . . . . . . 16 Ins4 SI3 I V
128127, 123inex 4105 . . . . . . . . . . . . . . 15 ( Ins4 SI3 I ∩ Ins2 Ins2 S ) V
129 1cex 4142 . . . . . . . . . . . . . . 15 1c V
130128, 129imaex 4747 . . . . . . . . . . . . . 14 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) V
131130ins4ex 5799 . . . . . . . . . . . . 13 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) V
132131ins2ex 5797 . . . . . . . . . . . 12 Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) V
133121ins3ex 5798 . . . . . . . . . . . . . . . . 17 Ins3 S V
134133ins2ex 5797 . . . . . . . . . . . . . . . 16 Ins2 Ins3 S V
135134ins2ex 5797 . . . . . . . . . . . . . . 15 Ins2 Ins2 Ins3 S V
136135ins2ex 5797 . . . . . . . . . . . . . 14 Ins2 Ins2 Ins2 Ins3 S V
137127, 136inex 4105 . . . . . . . . . . . . 13 ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) V
138137, 129imaex 4747 . . . . . . . . . . . 12 (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c) V
139132, 138symdifex 4108 . . . . . . . . . . 11 ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) V
140124, 139inex 4105 . . . . . . . . . 10 ( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) V
141140, 129imaex 4747 . . . . . . . . 9 (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) V
142141complex 4104 . . . . . . . 8 ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) V
143125ins3ex 5798 . . . . . . . 8 Ins3 I V
144142, 143difex 4107 . . . . . . 7 ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I ) V
145123, 144inex 4105 . . . . . 6 ( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) V
146145, 129imaex 4747 . . . . 5 (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c) V
147122, 146inex 4105 . . . 4 ( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) V
148147, 129imaex 4747 . . 3 (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c) V
149148complex 4104 . 2 ∼ (( Ins2 S ∩ (( Ins2 Ins2 S ∩ ( ∼ (( Ins2 Ins2 Ins2 S ∩ ( Ins2 Ins4 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ⊕ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins3 S ) “ 1c))) “ 1c) Ins3 I )) “ 1c)) “ 1c) V
150120, 149eqeltri 2423 1 Ext V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206   ∩ cin 3208   ⊕ csymdif 3209  {csn 3737  1cc1c 4134  ⟨cop 4561  {copab 4622   class class class wbr 4639   S csset 4719   “ cima 4722   I cid 4763   Ins2 cins2 5749   Ins3 cins3 5751   Ins4 cins4 5755   SI3 csi3 5757   Ext cext 5896 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-ext 5907 This theorem is referenced by: (None)
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