New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  addcfnex GIF version

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

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
 Copyright terms: Public domain W3C validator