Theorem addcfnex 5825

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.

Step	Hyp	Ref	Expression
1		df-addcfn 5747	. . 3 ⊢ AddC = (x ∈ V, y ∈ V ↦ (x +c y))
2		elin 3220	. . . . . . . 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 4112	. . . . . . . . . . 11 ⊢ {z} ∈ V
4	3	otelins2 5792	. . . . . . . . . 10 ⊢ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{b}, ⟨x, y⟩⟩ ∈ Ins2 S )
5		vex 2863	. . . . . . . . . . 11 ⊢ x ∈ V
6	5	otelins2 5792	. . . . . . . . . 10 ⊢ (⟨{b}, ⟨x, y⟩⟩ ∈ Ins2 S ↔ ⟨{b}, y⟩ ∈ S )
7		vex 2863	. . . . . . . . . . 11 ⊢ b ∈ V
8		vex 2863	. . . . . . . . . . 11 ⊢ y ∈ V
9	7, 8	opelssetsn 4761	. . . . . . . . . 10 ⊢ (⟨{b}, y⟩ ∈ S ↔ b ∈ y)
10	4, 6, 9	3bitri 262	. . . . . . . . 9 ⊢ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins2 Ins2 S ↔ b ∈ y)
11	8	oqelins4 5795	. . . . . . . . . 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 3220	. . . . . . . . . . . . 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 4112	. . . . . . . . . . . . . . . 16 ⊢ {b} ∈ V
14	13	otelins2 5792	. . . . . . . . . . . . . . 15 ⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{a}, ⟨{z}, x⟩⟩ ∈ Ins2 S )
15	3	otelins2 5792	. . . . . . . . . . . . . . 15 ⊢ (⟨{a}, ⟨{z}, x⟩⟩ ∈ Ins2 S ↔ ⟨{a}, x⟩ ∈ S )
16		vex 2863	. . . . . . . . . . . . . . . 16 ⊢ a ∈ V
17	16, 5	opelssetsn 4761	. . . . . . . . . . . . . . 15 ⊢ (⟨{a}, x⟩ ∈ S ↔ a ∈ x)
18	14, 15, 17	3bitri 262	. . . . . . . . . . . . . 14 ⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins2 Ins2 S ↔ a ∈ x)
19	5	oqelins4 5795	. . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . . . 16 ⊢ z ∈ V
21	16, 7, 20	otsnelsi3 5806	. . . . . . . . . . . . . . 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 3220	. . . . . . . . . . . . . . . 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 )))
23	20	otelins3 5793	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨a, ⟨b, z⟩⟩ ∈ Ins3 Disj ↔ ⟨a, b⟩ ∈ Disj )
24		df-br 4641	. . . . . . . . . . . . . . . . . 18 ⊢ (a Disj b ↔ ⟨a, b⟩ ∈ Disj )
25	16, 7	brdisj 5823	. . . . . . . . . . . . . . . . . 18 ⊢ (a Disj b ↔ (a ∩ b) = ∅)
26	23, 24, 25	3bitr2i 264	. . . . . . . . . . . . . . . . 17 ⊢ (⟨a, ⟨b, z⟩⟩ ∈ Ins3 Disj ↔ (a ∩ b) = ∅)
27		trtxp 5782	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (p((2nd ∘ 1st ) ⊗ 2nd )⟨b, z⟩ ↔ (p(2nd ∘ 1st )b ∧ p2nd z))
28	27	anbi2i 675	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((p(1st ∘ 1st )a ∧ p((2nd ∘ 1st ) ⊗ 2nd )⟨b, z⟩) ↔ (p(1st ∘ 1st )a ∧ (p(2nd ∘ 1st )b ∧ p2nd z)))
29		trtxp 5782	. . . . . . . . . . . . . . . . . . . . . 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)))
31	28, 29, 30	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, z⟩⟩ ↔ ((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ∧ p2nd z))
32		brco 4884	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (p(1st ∘ 1st )a ↔ ∃x(p1st x ∧ x1st a))
33	16	br1st 4859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (x1st a ↔ ∃y x = ⟨a, y⟩)
34	33	anbi2i 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⟩))
36	34, 35	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((p1st x ∧ x1st a) ↔ ∃y(p1st x ∧ x = ⟨a, y⟩))
37	36	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃x(p1st x ∧ x1st a) ↔ ∃x∃y(p1st x ∧ x = ⟨a, y⟩))
38		excom 1741	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃x∃y(p1st x ∧ x = ⟨a, y⟩) ↔ ∃y∃x(p1st x ∧ x = ⟨a, y⟩))
39		exancom 1586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∃x(p1st x ∧ x = ⟨a, y⟩) ↔ ∃x(x = ⟨a, y⟩ ∧ p1st x))
40	16, 8	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ⟨a, y⟩ ∈ V
41		breq2 4644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (x = ⟨a, y⟩ → (p1st x ↔ p1st ⟨a, y⟩))
42	40, 41	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∃x(x = ⟨a, y⟩ ∧ p1st x) ↔ p1st ⟨a, y⟩)
43	39, 42	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃x(p1st x ∧ x = ⟨a, y⟩) ↔ p1st ⟨a, y⟩)
44	43	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃y∃x(p1st x ∧ x = ⟨a, y⟩) ↔ ∃y p1st ⟨a, y⟩)
45	38, 44	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃x∃y(p1st x ∧ x = ⟨a, y⟩) ↔ ∃y p1st ⟨a, y⟩)
46	32, 37, 45	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (p(1st ∘ 1st )a ↔ ∃y p1st ⟨a, y⟩)
47	46	anbi1i 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))
49	40	br1st 4859	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (p1st ⟨a, y⟩ ↔ ∃z p = ⟨⟨a, y⟩, z⟩)
50		breq1 4643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (p = ⟨⟨a, y⟩, z⟩ → (p(2nd ∘ 1st )b ↔ ⟨⟨a, y⟩, z⟩(2nd ∘ 1st )b))
51	40, 20	brco1st 5778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨⟨a, y⟩, z⟩(2nd ∘ 1st )b ↔ ⟨a, y⟩2nd b)
52	16, 8	opbr2nd 5503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨a, y⟩2nd b ↔ y = b)
53	51, 52	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨⟨a, y⟩, z⟩(2nd ∘ 1st )b ↔ y = b)
54	50, 53	syl6bb 252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (p = ⟨⟨a, y⟩, z⟩ → (p(2nd ∘ 1st )b ↔ y = b))
55	54	exlimiv 1634	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃z p = ⟨⟨a, y⟩, z⟩ → (p(2nd ∘ 1st )b ↔ y = b))
56	49, 55	sylbi 187	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (p1st ⟨a, y⟩ → (p(2nd ∘ 1st )b ↔ y = b))
57	56	pm5.32i 618	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((p1st ⟨a, y⟩ ∧ p(2nd ∘ 1st )b) ↔ (p1st ⟨a, y⟩ ∧ y = b))
58	57	exbii 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⟩))
60	58, 59	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃y(p1st ⟨a, y⟩ ∧ p(2nd ∘ 1st )b) ↔ ∃y(y = b ∧ p1st ⟨a, y⟩))
61	47, 48, 60	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ↔ ∃y(y = b ∧ p1st ⟨a, y⟩))
62		opeq2 4580	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (y = b → ⟨a, y⟩ = ⟨a, b⟩)
63	62	breq2d 4652	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (y = b → (p1st ⟨a, y⟩ ↔ p1st ⟨a, b⟩))
64	7, 63	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃y(y = b ∧ p1st ⟨a, y⟩) ↔ p1st ⟨a, b⟩)
65	61, 64	bitri 240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ↔ p1st ⟨a, b⟩)
66	65	anbi1i 676	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((p(1st ∘ 1st )a ∧ p(2nd ∘ 1st )b) ∧ p2nd z) ↔ (p1st ⟨a, b⟩ ∧ p2nd z))
67	16, 7	opex 4589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨a, b⟩ ∈ V
68	67, 20	op1st2nd 5791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((p1st ⟨a, b⟩ ∧ p2nd z) ↔ p = ⟨⟨a, b⟩, z⟩)
69	31, 66, 68	3bitri 262	. . . . . . . . . . . . . . . . . . . 20 ⊢ (p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, z⟩⟩ ↔ p = ⟨⟨a, b⟩, z⟩)
70	69	rexbii 2640	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃p ∈ Cup p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, z⟩⟩ ↔ ∃p ∈ Cup p = ⟨⟨a, b⟩, z⟩)
71		elima 4755	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨a, ⟨b, z⟩⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ) ↔ ∃p ∈ Cup p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, z⟩⟩)
72		risset 2662	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨⟨a, b⟩, z⟩ ∈ Cup ↔ ∃p ∈ Cup p = ⟨⟨a, b⟩, z⟩)
73	70, 71, 72	3bitr4i 268	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨a, ⟨b, z⟩⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ) ↔ ⟨⟨a, b⟩, z⟩ ∈ Cup )
74		df-br 4641	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨a, b⟩ Cup z ↔ ⟨⟨a, b⟩, z⟩ ∈ Cup )
75	16, 7	brcup 5816	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨a, b⟩ Cup z ↔ z = (a ∪ b))
76	73, 74, 75	3bitr2i 264	. . . . . . . . . . . . . . . . 17 ⊢ (⟨a, ⟨b, z⟩⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ) ↔ z = (a ∪ b))
77	26, 76	anbi12i 678	. . . . . . . . . . . . . . . 16 ⊢ ((⟨a, ⟨b, z⟩⟩ ∈ Ins3 Disj ∧ ⟨a, ⟨b, z⟩⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup )) ↔ ((a ∩ b) = ∅ ∧ z = (a ∪ b)))
78	22, 77	bitri 240	. . . . . . . . . . . . . . 15 ⊢ (⟨a, ⟨b, z⟩⟩ ∈ ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup )) ↔ ((a ∩ b) = ∅ ∧ z = (a ∪ b)))
79	19, 21, 78	3bitri 262	. . . . . . . . . . . . . 14 ⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup )) ↔ ((a ∩ b) = ∅ ∧ z = (a ∪ b)))
80	18, 79	anbi12i 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 ∧ ((a ∩ b) = ∅ ∧ z = (a ∪ b))))
81	12, 80	bitri 240	. . . . . . . . . . . 12 ⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ))) ↔ (a ∈ x ∧ ((a ∩ b) = ∅ ∧ z = (a ∪ b))))
82	81	exbii 1582	. . . . . . . . . . 11 ⊢ (∃a⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ))) ↔ ∃a(a ∈ x ∧ ((a ∩ b) = ∅ ∧ z = (a ∪ b))))
83		elima1c 4948	. . . . . . . . . . 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 2621	. . . . . . . . . . 11 ⊢ (∃a ∈ x ((a ∩ b) = ∅ ∧ z = (a ∪ b)) ↔ ∃a(a ∈ x ∧ ((a ∩ b) = ∅ ∧ z = (a ∪ b))))
85	82, 83, 84	3bitr4i 268	. . . . . . . . . 10 ⊢ (⟨{b}, ⟨{z}, x⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) ↔ ∃a ∈ x ((a ∩ b) = ∅ ∧ z = (a ∪ b)))
86	11, 85	bitri 240	. . . . . . . . 9 ⊢ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) ↔ ∃a ∈ x ((a ∩ b) = ∅ ∧ z = (a ∪ b)))
87	10, 86	anbi12i 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 ((a ∩ b) = ∅ ∧ z = (a ∪ b))))
88	2, 87	bitri 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 ((a ∩ b) = ∅ ∧ z = (a ∪ b))))
89	88	exbii 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 ((a ∩ b) = ∅ ∧ z = (a ∪ b))))
90		elima1c 4948	. . . . . 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 2621	. . . . . 6 ⊢ (∃b ∈ y ∃a ∈ x ((a ∩ b) = ∅ ∧ z = (a ∪ b)) ↔ ∃b(b ∈ y ∧ ∃a ∈ x ((a ∩ b) = ∅ ∧ z = (a ∪ b))))
92	89, 90, 91	3bitr4i 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 ((a ∩ b) = ∅ ∧ z = (a ∪ b)))
93		eladdc 4399	. . . . . 6 ⊢ (z ∈ (x +c y) ↔ ∃a ∈ x ∃b ∈ y ((a ∩ b) = ∅ ∧ z = (a ∪ b)))
94		rexcom 2773	. . . . . 6 ⊢ (∃a ∈ x ∃b ∈ y ((a ∩ b) = ∅ ∧ z = (a ∪ b)) ↔ ∃b ∈ y ∃a ∈ x ((a ∩ b) = ∅ ∧ z = (a ∪ b)))
95	93, 94	bitri 240	. . . . 5 ⊢ (z ∈ (x +c y) ↔ ∃b ∈ y ∃a ∈ x ((a ∩ b) = ∅ ∧ z = (a ∪ b)))
96	92, 95	bitr4i 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))
97	96	releqmpt2 5810	. . 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))
98	1, 97	eqtr4i 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 4110	. . 3 ⊢ V ∈ V
100		ssetex 4745	. . . . . . 7 ⊢ S ∈ V
101	100	ins2ex 5798	. . . . . 6 ⊢ Ins2 S ∈ V
102	101	ins2ex 5798	. . . . 5 ⊢ Ins2 Ins2 S ∈ V
103		disjex 5824	. . . . . . . . . . . 12 ⊢ Disj ∈ V
104	103	ins3ex 5799	. . . . . . . . . . 11 ⊢ Ins3 Disj ∈ V
105		1stex 4740	. . . . . . . . . . . . . 14 ⊢ 1st ∈ V
106	105, 105	coex 4751	. . . . . . . . . . . . 13 ⊢ (1st ∘ 1st ) ∈ V
107		2ndex 5113	. . . . . . . . . . . . . . 15 ⊢ 2nd ∈ V
108	107, 105	coex 4751	. . . . . . . . . . . . . 14 ⊢ (2nd ∘ 1st ) ∈ V
109	108, 107	txpex 5786	. . . . . . . . . . . . 13 ⊢ ((2nd ∘ 1st ) ⊗ 2nd ) ∈ V
110	106, 109	txpex 5786	. . . . . . . . . . . 12 ⊢ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ∈ V
111		cupex 5817	. . . . . . . . . . . 12 ⊢ Cup ∈ V
112	110, 111	imaex 4748	. . . . . . . . . . 11 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ) ∈ V
113	104, 112	inex 4106	. . . . . . . . . 10 ⊢ ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup )) ∈ V
114	113	si3ex 5807	. . . . . . . . 9 ⊢ SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup )) ∈ V
115	114	ins4ex 5800	. . . . . . . 8 ⊢ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup )) ∈ V
116	102, 115	inex 4106	. . . . . . 7 ⊢ ( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ))) ∈ V
117		1cex 4143	. . . . . . 7 ⊢ 1c ∈ V
118	116, 117	imaex 4748	. . . . . 6 ⊢ (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) ∈ V
119	118	ins4ex 5800	. . . . 5 ⊢ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ))) “ 1c) ∈ V
120	102, 119	inex 4106	. . . 4 ⊢ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) ∈ V
121	120, 117	imaex 4748	. . 3 ⊢ (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins3 Disj ∩ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ Cup ))) “ 1c)) “ 1c) ∈ V
122	99, 99, 121	mpt2exlem 5812	. 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
123	98, 122	eqeltri 2423	1 ⊢ AddC ∈ V

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209   ⊕ csymdif 3210  ∅c0 3551  {csn 3738  1_cc1c 4135   +_c cplc 4376  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   S csset 4720   ∘ ccom 4722   “ cima 4723   × cxp 4771  2^nd c2nd 4784   ↦ cmpt2 5654   ⊗ ctxp 5736   Cup ccup 5742   Disj cdisj 5744   AddC caddcfn 5746   Ins2 cins2 5750   Ins3 cins3 5752   Ins4 cins4 5756   SI₃ csi3 5758

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-cup 5743  df-disj 5745  df-addcfn 5747  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759

This theorem is used by:  csucex  6260  addccan2nclem2  6265  nncdiv3lem2  6277  nnc3n3p1  6279  nchoicelem16  6305