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Theorem addcfnex 5824
Description: The cardinal addition function exists. (Contributed by SF, 12-Feb-2015.)
Assertion
Ref Expression
addcfnex AddC V

Proof of Theorem addcfnex
Dummy variables x y z a b p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-addcfn 5746 . . 3 AddC = (x V, y V (x +c y))
2 elin 3219 . . . . . . . 8 ({b}, {z}, x, y ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) ↔ ({b}, {z}, x, y Ins2 Ins2 S {b}, {z}, x, y Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)))
3 snex 4111 . . . . . . . . . . 11 {z} V
43otelins2 5791 . . . . . . . . . 10 ({b}, {z}, x, y Ins2 Ins2 S {b}, x, y Ins2 S )
5 vex 2862 . . . . . . . . . . 11 x V
65otelins2 5791 . . . . . . . . . 10 ({b}, x, y Ins2 S {b}, y S )
7 vex 2862 . . . . . . . . . . 11 b V
8 vex 2862 . . . . . . . . . . 11 y V
97, 8opelssetsn 4760 . . . . . . . . . 10 ({b}, y S b y)
104, 6, 93bitri 262 . . . . . . . . 9 ({b}, {z}, x, y Ins2 Ins2 S b y)
118oqelins4 5794 . . . . . . . . . 10 ({b}, {z}, x, y Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) ↔ {b}, {z}, x (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c))
12 elin 3219 . . . . . . . . . . . . 13 ({a}, {b}, {z}, x ( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) ↔ ({a}, {b}, {z}, x Ins2 Ins2 S {a}, {b}, {z}, x Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))))
13 snex 4111 . . . . . . . . . . . . . . . 16 {b} V
1413otelins2 5791 . . . . . . . . . . . . . . 15 ({a}, {b}, {z}, x Ins2 Ins2 S {a}, {z}, x Ins2 S )
153otelins2 5791 . . . . . . . . . . . . . . 15 ({a}, {z}, x Ins2 S {a}, x S )
16 vex 2862 . . . . . . . . . . . . . . . 16 a V
1716, 5opelssetsn 4760 . . . . . . . . . . . . . . 15 ({a}, x S a x)
1814, 15, 173bitri 262 . . . . . . . . . . . . . 14 ({a}, {b}, {z}, x Ins2 Ins2 S a x)
195oqelins4 5794 . . . . . . . . . . . . . . 15 ({a}, {b}, {z}, x Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )) ↔ {a}, {b}, {z} SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )))
20 vex 2862 . . . . . . . . . . . . . . . 16 z V
2116, 7, 20otsnelsi3 5805 . . . . . . . . . . . . . . 15 ({a}, {b}, {z} SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )) ↔ a, b, z ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )))
22 elin 3219 . . . . . . . . . . . . . . . 16 (a, b, z ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )) ↔ (a, b, z Ins3 Disj a, b, z (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )))
2320otelins3 5792 . . . . . . . . . . . . . . . . . 18 (a, b, z Ins3 Disja, b Disj )
24 df-br 4640 . . . . . . . . . . . . . . . . . 18 (a Disj ba, b Disj )
2516, 7brdisj 5822 . . . . . . . . . . . . . . . . . 18 (a Disj b ↔ (ab) = )
2623, 24, 253bitr2i 264 . . . . . . . . . . . . . . . . 17 (a, b, z Ins3 Disj ↔ (ab) = )
27 trtxp 5781 . . . . . . . . . . . . . . . . . . . . . . 23 (p((2nd 1st ) ⊗ 2nd )b, z ↔ (p(2nd 1st )b p2nd z))
2827anbi2i 675 . . . . . . . . . . . . . . . . . . . . . 22 ((p(1st 1st )a p((2nd 1st ) ⊗ 2nd )b, z) ↔ (p(1st 1st )a (p(2nd 1st )b p2nd z)))
29 trtxp 5781 . . . . . . . . . . . . . . . . . . . . . 22 (p((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd ))a, b, z ↔ (p(1st 1st )a p((2nd 1st ) ⊗ 2nd )b, z))
30 anass 630 . . . . . . . . . . . . . . . . . . . . . 22 (((p(1st 1st )a p(2nd 1st )b) p2nd z) ↔ (p(1st 1st )a (p(2nd 1st )b p2nd z)))
3128, 29, 303bitr4i 268 . . . . . . . . . . . . . . . . . . . . 21 (p((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd ))a, b, z ↔ ((p(1st 1st )a p(2nd 1st )b) p2nd z))
32 brco 4883 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (p(1st 1st )ax(p1st x x1st a))
3316br1st 4858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (x1st ay x = a, y)
3433anbi2i 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((p1st x x1st a) ↔ (p1st x y x = a, y))
35 19.42v 1905 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (y(p1st x x = a, y) ↔ (p1st x y x = a, y))
3634, 35bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((p1st x x1st a) ↔ y(p1st x x = a, y))
3736exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (x(p1st x x1st a) ↔ xy(p1st x x = a, y))
38 excom 1741 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (xy(p1st x x = a, y) ↔ yx(p1st x x = a, y))
39 exancom 1586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (x(p1st x x = a, y) ↔ x(x = a, y p1st x))
4016, 8opex 4588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 a, y V
41 breq2 4643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (x = a, y → (p1st xp1st a, y))
4240, 41ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (x(x = a, y p1st x) ↔ p1st a, y)
4339, 42bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (x(p1st x x = a, y) ↔ p1st a, y)
4443exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (yx(p1st x x = a, y) ↔ y p1st a, y)
4538, 44bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (xy(p1st x x = a, y) ↔ y p1st a, y)
4632, 37, 453bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . 25 (p(1st 1st )ay p1st a, y)
4746anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . 24 ((p(1st 1st )a p(2nd 1st )b) ↔ (y p1st a, y p(2nd 1st )b))
48 19.41v 1901 . . . . . . . . . . . . . . . . . . . . . . . 24 (y(p1st a, y p(2nd 1st )b) ↔ (y p1st a, y p(2nd 1st )b))
4940br1st 4858 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (p1st a, yz p = a, y, z)
50 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (p = a, y, z → (p(2nd 1st )ba, y, z(2nd 1st )b))
5140, 20brco1st 5777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (a, y, z(2nd 1st )ba, y2nd b)
5216, 8opbr2nd 5502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (a, y2nd by = b)
5351, 52bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (a, y, z(2nd 1st )by = b)
5450, 53syl6bb 252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (p = a, y, z → (p(2nd 1st )by = b))
5554exlimiv 1634 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (z p = a, y, z → (p(2nd 1st )by = b))
5649, 55sylbi 187 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (p1st a, y → (p(2nd 1st )by = b))
5756pm5.32i 618 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((p1st a, y p(2nd 1st )b) ↔ (p1st a, y y = b))
5857exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . 25 (y(p1st a, y p(2nd 1st )b) ↔ y(p1st a, y y = b))
59 exancom 1586 . . . . . . . . . . . . . . . . . . . . . . . . 25 (y(p1st a, y y = b) ↔ y(y = b p1st a, y))
6058, 59bitri 240 . . . . . . . . . . . . . . . . . . . . . . . 24 (y(p1st a, y p(2nd 1st )b) ↔ y(y = b p1st a, y))
6147, 48, 603bitr2i 264 . . . . . . . . . . . . . . . . . . . . . . 23 ((p(1st 1st )a p(2nd 1st )b) ↔ y(y = b p1st a, y))
62 opeq2 4579 . . . . . . . . . . . . . . . . . . . . . . . . 25 (y = ba, y = a, b)
6362breq2d 4651 . . . . . . . . . . . . . . . . . . . . . . . 24 (y = b → (p1st a, yp1st a, b))
647, 63ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . 23 (y(y = b p1st a, y) ↔ p1st a, b)
6561, 64bitri 240 . . . . . . . . . . . . . . . . . . . . . 22 ((p(1st 1st )a p(2nd 1st )b) ↔ p1st a, b)
6665anbi1i 676 . . . . . . . . . . . . . . . . . . . . 21 (((p(1st 1st )a p(2nd 1st )b) p2nd z) ↔ (p1st a, b p2nd z))
6716, 7opex 4588 . . . . . . . . . . . . . . . . . . . . . 22 a, b V
6867, 20op1st2nd 5790 . . . . . . . . . . . . . . . . . . . . 21 ((p1st a, b p2nd z) ↔ p = a, b, z)
6931, 66, 683bitri 262 . . . . . . . . . . . . . . . . . . . 20 (p((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd ))a, b, zp = a, b, z)
7069rexbii 2639 . . . . . . . . . . . . . . . . . . 19 (p Cup p((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd ))a, b, zp Cup p = a, b, z)
71 elima 4754 . . . . . . . . . . . . . . . . . . 19 (a, b, z (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ) ↔ p Cup p((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd ))a, b, z)
72 risset 2661 . . . . . . . . . . . . . . . . . . 19 (a, b, z Cupp Cup p = a, b, z)
7370, 71, 723bitr4i 268 . . . . . . . . . . . . . . . . . 18 (a, b, z (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ) ↔ a, b, z Cup )
74 df-br 4640 . . . . . . . . . . . . . . . . . 18 (a, b Cup za, b, z Cup )
7516, 7brcup 5815 . . . . . . . . . . . . . . . . . 18 (a, b Cup zz = (ab))
7673, 74, 753bitr2i 264 . . . . . . . . . . . . . . . . 17 (a, b, z (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ) ↔ z = (ab))
7726, 76anbi12i 678 . . . . . . . . . . . . . . . 16 ((a, b, z Ins3 Disj a, b, z (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )) ↔ ((ab) = z = (ab)))
7822, 77bitri 240 . . . . . . . . . . . . . . 15 (a, b, z ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )) ↔ ((ab) = z = (ab)))
7919, 21, 783bitri 262 . . . . . . . . . . . . . 14 ({a}, {b}, {z}, x Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )) ↔ ((ab) = z = (ab)))
8018, 79anbi12i 678 . . . . . . . . . . . . 13 (({a}, {b}, {z}, x Ins2 Ins2 S {a}, {b}, {z}, x Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) ↔ (a x ((ab) = z = (ab))))
8112, 80bitri 240 . . . . . . . . . . . 12 ({a}, {b}, {z}, x ( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) ↔ (a x ((ab) = z = (ab))))
8281exbii 1582 . . . . . . . . . . 11 (a{a}, {b}, {z}, x ( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) ↔ a(a x ((ab) = z = (ab))))
83 elima1c 4947 . . . . . . . . . . 11 ({b}, {z}, x (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) ↔ a{a}, {b}, {z}, x ( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))))
84 df-rex 2620 . . . . . . . . . . 11 (a x ((ab) = z = (ab)) ↔ a(a x ((ab) = z = (ab))))
8582, 83, 843bitr4i 268 . . . . . . . . . 10 ({b}, {z}, x (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) ↔ a x ((ab) = z = (ab)))
8611, 85bitri 240 . . . . . . . . 9 ({b}, {z}, x, y Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) ↔ a x ((ab) = z = (ab)))
8710, 86anbi12i 678 . . . . . . . 8 (({b}, {z}, x, y Ins2 Ins2 S {b}, {z}, x, y Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) ↔ (b y a x ((ab) = z = (ab))))
882, 87bitri 240 . . . . . . 7 ({b}, {z}, x, y ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) ↔ (b y a x ((ab) = z = (ab))))
8988exbii 1582 . . . . . 6 (b{b}, {z}, x, y ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) ↔ b(b y a x ((ab) = z = (ab))))
90 elima1c 4947 . . . . . 6 ({z}, x, y (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) “ 1c) ↔ b{b}, {z}, x, y ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)))
91 df-rex 2620 . . . . . 6 (b y a x ((ab) = z = (ab)) ↔ b(b y a x ((ab) = z = (ab))))
9289, 90, 913bitr4i 268 . . . . 5 ({z}, x, y (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) “ 1c) ↔ b y a x ((ab) = z = (ab)))
93 eladdc 4398 . . . . . 6 (z (x +c y) ↔ a x b y ((ab) = z = (ab)))
94 rexcom 2772 . . . . . 6 (a x b y ((ab) = z = (ab)) ↔ b y a x ((ab) = z = (ab)))
9593, 94bitri 240 . . . . 5 (z (x +c y) ↔ b y a x ((ab) = z = (ab)))
9692, 95bitr4i 243 . . . 4 ({z}, x, y (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) “ 1c) ↔ z (x +c y))
9796releqmpt2 5809 . . 3 (((V × V) × V) (( Ins2 S Ins3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) “ 1c)) “ 1c)) = (x V, y V (x +c y))
981, 97eqtr4i 2376 . 2 AddC = (((V × V) × V) (( Ins2 S Ins3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) “ 1c)) “ 1c))
99 vvex 4109 . . 3 V V
100 ssetex 4744 . . . . . . 7 S V
101100ins2ex 5797 . . . . . 6 Ins2 S V
102101ins2ex 5797 . . . . 5 Ins2 Ins2 S V
103 disjex 5823 . . . . . . . . . . . 12 Disj V
104103ins3ex 5798 . . . . . . . . . . 11 Ins3 Disj V
105 1stex 4739 . . . . . . . . . . . . . 14 1st V
106105, 105coex 4750 . . . . . . . . . . . . 13 (1st 1st ) V
107 2ndex 5112 . . . . . . . . . . . . . . 15 2nd V
108107, 105coex 4750 . . . . . . . . . . . . . 14 (2nd 1st ) V
109108, 107txpex 5785 . . . . . . . . . . . . 13 ((2nd 1st ) ⊗ 2nd ) V
110106, 109txpex 5785 . . . . . . . . . . . 12 ((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) V
111 cupex 5816 . . . . . . . . . . . 12 Cup V
112110, 111imaex 4747 . . . . . . . . . . 11 (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ) V
113104, 112inex 4105 . . . . . . . . . 10 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )) V
114113si3ex 5806 . . . . . . . . 9 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )) V
115114ins4ex 5799 . . . . . . . 8 Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup )) V
116102, 115inex 4105 . . . . . . 7 ( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) V
117 1cex 4142 . . . . . . 7 1c V
118116, 117imaex 4747 . . . . . 6 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) V
119118ins4ex 5799 . . . . 5 Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) V
120102, 119inex 4105 . . . 4 ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) V
121120, 117imaex 4747 . . 3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) “ 1c) V
12299, 99, 121mpt2exlem 5811 . 2 (((V × V) × V) (( Ins2 S Ins3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ( Ins3 Disj ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) “ 1c)) “ 1c)) V
12398, 122eqeltri 2423 1 AddC V
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358  wex 1541   = wceq 1642   wcel 1710  wrex 2615  Vcvv 2859   cdif 3206  cun 3207  cin 3208  csymdif 3209  c0 3550  {csn 3737  1cc1c 4134   +c cplc 4375  cop 4561   class class class wbr 4639  1st c1st 4717   S csset 4719   ccom 4721  cima 4722   × cxp 4770  2nd c2nd 4783   cmpt2 5653  ctxp 5735   Cup ccup 5741   Disj cdisj 5743   AddC caddcfn 5745   Ins2 cins2 5749   Ins3 cins3 5751   Ins4 cins4 5755   SI3 csi3 5757
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-csb 3137  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-iun 3971  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-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758
This theorem is referenced by:  csucex  6259  addccan2nclem2  6264  nncdiv3lem2  6276  nnc3n3p1  6278  nchoicelem16  6304
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