Theorem relop 5614

Description: A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
relop.1	⊢ 𝐴 ∈ V
relop.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
relop	⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem relop

Dummy variables 𝑤 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rel 5457	. 2 ⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ ⊆ (V × V))
2		dfss2 3883	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ ∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)))
3		relop.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
4		relop.2	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
5	3, 4	elop 5258	. . . . . . . . 9 ⊢ (𝑧 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}))
6		elvv 5519	. . . . . . . . 9 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
7	5, 6	imbi12i 352	. . . . . . . 8 ⊢ ((𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
8		jaob 956	. . . . . . . 8 ⊢ (((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
9	7, 8	bitri 276	. . . . . . 7 ⊢ ((𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
10	9	albii 1805	. . . . . 6 ⊢ (∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ∀𝑧((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
11		19.26 1856	. . . . . 6 ⊢ (∀𝑧((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
12	10, 11	bitri 276	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
13	2, 12	bitri 276	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
14		snex 5230	. . . . . . 7 ⊢ {𝐴} ∈ V
15		eqeq1 2801	. . . . . . . 8 ⊢ (𝑧 = {𝐴} → (𝑧 = {𝐴} ↔ {𝐴} = {𝐴}))
16		eqeq1 2801	. . . . . . . . . 10 ⊢ (𝑧 = {𝐴} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ {𝐴} = ⟨𝑥, 𝑦⟩))
17		eqcom 2804	. . . . . . . . . . 11 ⊢ ({𝐴} = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = {𝐴})
18		vex 3443	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
19		vex 3443	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
20	18, 19	opeqsn 5292	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = {𝐴} ↔ (𝑥 = 𝑦 ∧ 𝐴 = {𝑥}))
21	17, 20	bitri 276	. . . . . . . . . 10 ⊢ ({𝐴} = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑦 ∧ 𝐴 = {𝑥}))
22	16, 21	syl6bb 288	. . . . . . . . 9 ⊢ (𝑧 = {𝐴} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑦 ∧ 𝐴 = {𝑥})))
23	22	2exbidv 1906	. . . . . . . 8 ⊢ (𝑧 = {𝐴} → (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥})))
24	15, 23	imbi12d 346	. . . . . . 7 ⊢ (𝑧 = {𝐴} → ((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ({𝐴} = {𝐴} → ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}))))
25	14, 24	spcv 3550	. . . . . 6 ⊢ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ({𝐴} = {𝐴} → ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥})))
26		sneq 4488	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤} = {𝑥})
27	26	eqeq2d 2807	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝐴 = {𝑤} ↔ 𝐴 = {𝑥}))
28	27	cbvexvw 2025	. . . . . . 7 ⊢ (∃𝑤 𝐴 = {𝑤} ↔ ∃𝑥 𝐴 = {𝑥})
29		ax6evr 2003	. . . . . . . . 9 ⊢ ∃𝑦 𝑥 = 𝑦
30		19.41v 1931	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}) ↔ (∃𝑦 𝑥 = 𝑦 ∧ 𝐴 = {𝑥}))
31	29, 30	mpbiran 705	. . . . . . . 8 ⊢ (∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}) ↔ 𝐴 = {𝑥})
32	31	exbii 1833	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}) ↔ ∃𝑥 𝐴 = {𝑥})
33		eqid 2797	. . . . . . . 8 ⊢ {𝐴} = {𝐴}
34	33	a1bi 364	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}) ↔ ({𝐴} = {𝐴} → ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥})))
35	28, 32, 34	3bitr2ri 301	. . . . . 6 ⊢ (({𝐴} = {𝐴} → ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥})) ↔ ∃𝑤 𝐴 = {𝑤})
36	25, 35	sylib 219	. . . . 5 ⊢ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∃𝑤 𝐴 = {𝑤})
37		eqid 2797	. . . . . 6 ⊢ {𝐴, 𝐵} = {𝐴, 𝐵}
38		prex 5231	. . . . . . 7 ⊢ {𝐴, 𝐵} ∈ V
39		eqeq1 2801	. . . . . . . 8 ⊢ (𝑧 = {𝐴, 𝐵} → (𝑧 = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴, 𝐵}))
40		eqeq1 2801	. . . . . . . . 9 ⊢ (𝑧 = {𝐴, 𝐵} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ {𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
41	40	2exbidv 1906	. . . . . . . 8 ⊢ (𝑧 = {𝐴, 𝐵} → (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
42	39, 41	imbi12d 346	. . . . . . 7 ⊢ (𝑧 = {𝐴, 𝐵} → ((𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ({𝐴, 𝐵} = {𝐴, 𝐵} → ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩)))
43	38, 42	spcv 3550	. . . . . 6 ⊢ (∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ({𝐴, 𝐵} = {𝐴, 𝐵} → ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
44	37, 43	mpi 20	. . . . 5 ⊢ (∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩)
45		eqcom 2804	. . . . . . . . . 10 ⊢ ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = {𝐴, 𝐵})
46	18, 19, 3, 4	opeqpr 5293	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = {𝐴, 𝐵} ↔ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})))
47	45, 46	bitri 276	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ ↔ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})))
48		idd 24	. . . . . . . . . 10 ⊢ (𝐴 = {𝑤} → ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
49		eqtr2 2819	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → {𝑥, 𝑦} = {𝑤})
50	18, 19	preqsn 4705	. . . . . . . . . . . . . . 15 ⊢ ({𝑥, 𝑦} = {𝑤} ↔ (𝑥 = 𝑦 ∧ 𝑦 = 𝑤))
51	50	simplbi 498	. . . . . . . . . . . . . 14 ⊢ ({𝑥, 𝑦} = {𝑤} → 𝑥 = 𝑦)
52	49, 51	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → 𝑥 = 𝑦)
53		dfsn2 4491	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑥} = {𝑥, 𝑥}
54		preq2 4583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑥} = {𝑥, 𝑦})
55	53, 54	syl5req 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑦} = {𝑥})
56	55	eqeq2d 2807	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐴 = {𝑥, 𝑦} ↔ 𝐴 = {𝑥}))
57	53, 54	syl5eq 2845	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑥, 𝑦})
58	57	eqeq2d 2807	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐵 = {𝑥} ↔ 𝐵 = {𝑥, 𝑦}))
59	56, 58	anbi12d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) ↔ (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
60	59	biimpd 230	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
61	60	expd 416	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐴 = {𝑥, 𝑦} → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
62	61	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝐴 = {𝑥, 𝑦} → (𝑥 = 𝑦 → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
63	62	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → (𝑥 = 𝑦 → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
64	52, 63	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
65	64	expcom 414	. . . . . . . . . . 11 ⊢ (𝐴 = {𝑤} → (𝐴 = {𝑥, 𝑦} → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
66	65	impd 411	. . . . . . . . . 10 ⊢ (𝐴 = {𝑤} → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
67	48, 66	jaod 854	. . . . . . . . 9 ⊢ (𝐴 = {𝑤} → (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
68	47, 67	syl5bi 243	. . . . . . . 8 ⊢ (𝐴 = {𝑤} → ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
69	68	2eximdv 1901	. . . . . . 7 ⊢ (𝐴 = {𝑤} → (∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
70	69	exlimiv 1912	. . . . . 6 ⊢ (∃𝑤 𝐴 = {𝑤} → (∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
71	70	imp 407	. . . . 5 ⊢ ((∃𝑤 𝐴 = {𝑤} ∧ ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩) → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
72	36, 44, 71	syl2an 595	. . . 4 ⊢ ((∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)) → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
73	13, 72	sylbi 218	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ⊆ (V × V) → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
74		simpr 485	. . . . . . . . . . 11 ⊢ ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → 𝑧 = {𝐴})
75		equid 2000	. . . . . . . . . . . . . 14 ⊢ 𝑥 = 𝑥
76	75	jctl 524	. . . . . . . . . . . . 13 ⊢ (𝐴 = {𝑥} → (𝑥 = 𝑥 ∧ 𝐴 = {𝑥}))
77	18, 18	opeqsn 5292	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑥⟩ = {𝐴} ↔ (𝑥 = 𝑥 ∧ 𝐴 = {𝑥}))
78	76, 77	sylibr 235	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥} → ⟨𝑥, 𝑥⟩ = {𝐴})
79	78	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → ⟨𝑥, 𝑥⟩ = {𝐴})
80	74, 79	eqtr4d 2836	. . . . . . . . . 10 ⊢ ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → 𝑧 = ⟨𝑥, 𝑥⟩)
81		opeq12 4718	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑥) → ⟨𝑤, 𝑣⟩ = ⟨𝑥, 𝑥⟩)
82	81	eqeq2d 2807	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑥) → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑥, 𝑥⟩))
83	18, 18, 82	spc2ev 3552	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑥⟩ → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
84	80, 83	syl 17	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
85	84	adantlr 711	. . . . . . . 8 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴}) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
86		preq12 4584	. . . . . . . . . . . 12 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → {𝐴, 𝐵} = {{𝑥}, {𝑥, 𝑦}})
87	86	eqeq2d 2807	. . . . . . . . . . 11 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝑧 = {𝐴, 𝐵} ↔ 𝑧 = {{𝑥}, {𝑥, 𝑦}}))
88	87	biimpa 477	. . . . . . . . . 10 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → 𝑧 = {{𝑥}, {𝑥, 𝑦}})
89	18, 19	dfop 4715	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
90	88, 89	syl6eqr 2851	. . . . . . . . 9 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → 𝑧 = ⟨𝑥, 𝑦⟩)
91		opeq12 4718	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑤, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
92	91	eqeq2d 2807	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
93	18, 19, 92	spc2ev 3552	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
94	90, 93	syl 17	. . . . . . . 8 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
95	85, 94	jaodan 952	. . . . . . 7 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ (𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵})) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
96	95	ex 413	. . . . . 6 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩))
97		elvv 5519	. . . . . 6 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
98	96, 5, 97	3imtr4g 297	. . . . 5 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)))
99	98	ssrdv 3901	. . . 4 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ⟨𝐴, 𝐵⟩ ⊆ (V × V))
100	99	exlimivv 1914	. . 3 ⊢ (∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ⟨𝐴, 𝐵⟩ ⊆ (V × V))
101	73, 100	impbii 210	. 2 ⊢ (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
102	1, 101	bitri 276	1 ⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842  ∀wal 1523   = wceq 1525  ∃wex 1765   ∈ wcel 2083  Vcvv 3440   ⊆ wss 3865  {csn 4478  {cpr 4480  ⟨cop 4484   × cxp 5448  Rel wrel 5455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-opab 5031  df-xp 5456  df-rel 5457

This theorem is referenced by:  funopg  6266