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Theorem connexex 5913
 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.
StepHypRef Expression
1 df-connex 5903 . . 3 Connex = {r, a x a y a (xry yrx)}
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 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 4947 . . . . . . . 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 3219 . . . . . . . . . 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)))
82otelins2 5791 . . . . . . . . . . . 12 ({x}, r, a Ins2 S {x}, a S )
9 vex 2862 . . . . . . . . . . . . 13 x V
109, 3opelssetsn 4760 . . . . . . . . . . . 12 ({x}, a S x a)
118, 10bitri 240 . . . . . . . . . . 11 ({x}, r, a Ins2 S x a)
12 elima1c 4947 . . . . . . . . . . . . 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 3221 . . . . . . . . . . . . . . 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 4111 . . . . . . . . . . . . . . . . . 18 {x} V
1514otelins2 5791 . . . . . . . . . . . . . . . . 17 ({y}, {x}, r, a Ins2 Ins2 S {y}, r, a Ins2 S )
162otelins2 5791 . . . . . . . . . . . . . . . . 17 ({y}, r, a Ins2 S {y}, a S )
17 vex 2862 . . . . . . . . . . . . . . . . . 18 y V
1817, 3opelssetsn 4760 . . . . . . . . . . . . . . . . 17 ({y}, a S y a)
1915, 16, 183bitri 262 . . . . . . . . . . . . . . . 16 ({y}, {x}, r, a Ins2 Ins2 S y a)
203oqelins4 5794 . . . . . . . . . . . . . . . . . 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 3220 . . . . . . . . . . . . . . . . . 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 3219 . . . . . . . . . . . . . . . . . . . . . 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 ))
232oqelins4 5794 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, {y}, {x}, r Ins4 SI3 Swap {p}, {y}, {x} SI3 Swap )
24 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . 25 p V
2524, 17, 9otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, {y}, {x} SI3 Swap p, y, x Swap )
26 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . 25 (p Swap y, xp, y, x Swap )
2717, 9brswap2 4860 . . . . . . . . . . . . . . . . . . . . . . . . 25 (p Swap y, xp = x, y)
2826, 27bitr3i 242 . . . . . . . . . . . . . . . . . . . . . . . 24 (p, y, x Swap p = x, y)
2923, 25, 283bitri 262 . . . . . . . . . . . . . . . . . . . . . . 23 ({p}, {y}, {x}, r Ins4 SI3 Swap p = x, y)
30 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . 25 {y} V
3130otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, {y}, {x}, r Ins2 Ins2 S {p}, {x}, r Ins2 S )
3214otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, {x}, r Ins2 S {p}, r S )
3324, 2opelssetsn 4760 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, r S p r)
3431, 32, 333bitri 262 . . . . . . . . . . . . . . . . . . . . . . 23 ({p}, {y}, {x}, r Ins2 Ins2 S p r)
3529, 34anbi12i 678 . . . . . . . . . . . . . . . . . . . . . 22 (({p}, {y}, {x}, r Ins4 SI3 Swap {p}, {y}, {x}, r Ins2 Ins2 S ) ↔ (p = x, y p r))
3622, 35bitri 240 . . . . . . . . . . . . . . . . . . . . 21 ({p}, {y}, {x}, r ( Ins4 SI3 Swap Ins2 Ins2 S ) ↔ (p = x, y p r))
3736exbii 1582 . . . . . . . . . . . . . . . . . . . 20 (p{p}, {y}, {x}, r ( Ins4 SI3 Swap Ins2 Ins2 S ) ↔ p(p = x, y p r))
38 elima1c 4947 . . . . . . . . . . . . . . . . . . . 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 4640 . . . . . . . . . . . . . . . . . . . . 21 (xryx, y r)
40 df-clel 2349 . . . . . . . . . . . . . . . . . . . . 21 (x, y rp(p = x, y p r))
4139, 40bitri 240 . . . . . . . . . . . . . . . . . . . 20 (xryp(p = x, y p r))
4237, 38, 413bitr4i 268 . . . . . . . . . . . . . . . . . . 19 ({y}, {x}, r (( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ↔ xry)
43 elin 3219 . . . . . . . . . . . . . . . . . . . . . 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 ))
442oqelins4 5794 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, {y}, {x}, r Ins4 SI3 I ↔ {p}, {y}, {x} SI3 I )
4524, 17, 9otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . . . . 24 ({p}, {y}, {x} SI3 I ↔ p, y, x I )
46 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . 25 (p I y, xp, y, x I )
4717, 9opex 4588 . . . . . . . . . . . . . . . . . . . . . . . . . 26 y, x V
4847ideq 4870 . . . . . . . . . . . . . . . . . . . . . . . . 25 (p I y, xp = y, x)
4946, 48bitr3i 242 . . . . . . . . . . . . . . . . . . . . . . . 24 (p, y, x I ↔ p = y, x)
5044, 45, 493bitri 262 . . . . . . . . . . . . . . . . . . . . . . 23 ({p}, {y}, {x}, r Ins4 SI3 I ↔ p = y, x)
5150, 34anbi12i 678 . . . . . . . . . . . . . . . . . . . . . 22 (({p}, {y}, {x}, r Ins4 SI3 I {p}, {y}, {x}, r Ins2 Ins2 S ) ↔ (p = y, x p r))
5243, 51bitri 240 . . . . . . . . . . . . . . . . . . . . 21 ({p}, {y}, {x}, r ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ (p = y, x p r))
5352exbii 1582 . . . . . . . . . . . . . . . . . . . 20 (p{p}, {y}, {x}, r ( Ins4 SI3 I ∩ Ins2 Ins2 S ) ↔ p(p = y, x p r))
54 elima1c 4947 . . . . . . . . . . . . . . . . . . . 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 4640 . . . . . . . . . . . . . . . . . . . . 21 (yrxy, x r)
56 df-clel 2349 . . . . . . . . . . . . . . . . . . . . 21 (y, x rp(p = y, x p r))
5755, 56bitri 240 . . . . . . . . . . . . . . . . . . . 20 (yrxp(p = y, x p r))
5853, 54, 573bitr4i 268 . . . . . . . . . . . . . . . . . . 19 ({y}, {x}, r (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) ↔ yrx)
5942, 58orbi12i 507 . . . . . . . . . . . . . . . . . 18 (({y}, {x}, r (( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) {y}, {x}, r (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (xry yrx))
6020, 21, 593bitri 262 . . . . . . . . . . . . . . . . 17 ({y}, {x}, r, a Ins4 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ (xry yrx))
6160notbii 287 . . . . . . . . . . . . . . . 16 {y}, {x}, r, a Ins4 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) ↔ ¬ (xry yrx))
6219, 61anbi12i 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)))
6313, 62bitri 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)))
6463exbii 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)))
6512, 64bitri 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 2620 . . . . . . . . . . . 12 (y a ¬ (xry yrx) ↔ y(y a ¬ (xry yrx)))
67 rexnal 2625 . . . . . . . . . . . 12 (y a ¬ (xry yrx) ↔ ¬ y a (xry yrx))
6865, 66, 673bitr2i 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))
6911, 68anbi12i 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)))
707, 69bitri 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)))
7170exbii 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)))
726, 71bitri 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 2620 . . . . . . 7 (x a ¬ y a (xry yrx) ↔ x(x a ¬ y a (xry yrx)))
74 rexnal 2625 . . . . . . 7 (x a ¬ y a (xry yrx) ↔ ¬ x a y a (xry yrx))
7572, 73, 743bitr2i 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))
7675con2bii 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))
775, 76bitr4i 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))
7877opabbi2i 4866 . . 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)}
791, 78eqtr4i 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 4744 . . . . . 6 S V
8180ins2ex 5797 . . . . 5 Ins2 S V
8281ins2ex 5797 . . . . . . 7 Ins2 Ins2 S V
83 swapex 4742 . . . . . . . . . . . . 13 Swap V
8483si3ex 5806 . . . . . . . . . . . 12 SI3 Swap V
8584ins4ex 5799 . . . . . . . . . . 11 Ins4 SI3 Swap V
8685, 82inex 4105 . . . . . . . . . 10 ( Ins4 SI3 Swap Ins2 Ins2 S ) V
87 1cex 4142 . . . . . . . . . 10 1c V
8886, 87imaex 4747 . . . . . . . . 9 (( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) V
89 idex 5504 . . . . . . . . . . . . 13 I V
9089si3ex 5806 . . . . . . . . . . . 12 SI3 I V
9190ins4ex 5799 . . . . . . . . . . 11 Ins4 SI3 I V
9291, 82inex 4105 . . . . . . . . . 10 ( Ins4 SI3 I ∩ Ins2 Ins2 S ) V
9392, 87imaex 4747 . . . . . . . . 9 (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c) V
9488, 93unex 4106 . . . . . . . 8 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) V
9594ins4ex 5799 . . . . . . 7 Ins4 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c)) V
9682, 95difex 4107 . . . . . 6 ( Ins2 Ins2 S Ins4 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) V
9796, 87imaex 4747 . . . . 5 (( Ins2 Ins2 S Ins4 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c) V
9881, 97inex 4105 . . . 4 ( Ins2 S ∩ (( Ins2 Ins2 S Ins4 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) V
9998, 87imaex 4747 . . 3 (( Ins2 S ∩ (( Ins2 Ins2 S Ins4 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) V
10099complex 4104 . 2 ∼ (( Ins2 S ∩ (( Ins2 Ins2 S Ins4 ((( Ins4 SI3 Swap Ins2 Ins2 S ) “ 1c) ∪ (( Ins4 SI3 I ∩ Ins2 Ins2 S ) “ 1c))) “ 1c)) “ 1c) V
10179, 100eqeltri 2423 1 Connex V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  {csn 3737  1cc1c 4134  ⟨cop 4561  {copab 4622   class class class wbr 4639   Swap cswap 4718   S csset 4719   “ cima 4722   I cid 4763   Ins2 cins2 5749   Ins4 cins4 5755   SI3 csi3 5757   Connex cconnex 5892 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-ins4 5756  df-si3 5758  df-connex 5903 This theorem is referenced by:  strictex  5918
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