Theorem elvvv 5662

Description: Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)

Assertion

Ref	Expression
elvvv	⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem elvvv

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 5612	. 2 ⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑤∃𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
2		ancom 461	. . . . . 6 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
3	2	2exbii 1851	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
4		19.42vv 1961	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
5		elvv 5661	. . . . . . 7 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
6	5	anbi2i 623	. . . . . 6 ⊢ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
7		vex 3436	. . . . . . 7 ⊢ 𝑧 ∈ V
8	7	biantru 530	. . . . . 6 ⊢ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V))
9	4, 6, 8	3bitr2i 299	. . . . 5 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V))
10		anass 469	. . . . 5 ⊢ (((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
11	3, 9, 10	3bitrri 298	. . . 4 ⊢ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
12	11	2exbii 1851	. . 3 ⊢ (∃𝑤∃𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑤∃𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
13		exrot4 2166	. . 3 ⊢ (∃𝑥∃𝑦∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑤∃𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
14		excom 2162	. . . . 5 ⊢ (∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
15		opex 5379	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
16		opeq1 4804	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
17	16	eqeq2d 2749	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑤, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
18	15, 17	ceqsexv 3479	. . . . . 6 ⊢ (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
19	18	exbii 1850	. . . . 5 ⊢ (∃𝑧∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
20	14, 19	bitri 274	. . . 4 ⊢ (∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
21	20	2exbii 1851	. . 3 ⊢ (∃𝑥∃𝑦∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
22	12, 13, 21	3bitr2i 299	. 2 ⊢ (∃𝑤∃𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
23	1, 22	bitri 274	1 ⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   × cxp 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595

This theorem is referenced by:  ssrelrel  5706  dftpos3  8060