Theorem elvvv 5738

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 5685	. 2 ⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑤∃𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
2		ancom 465	. . . . . 6 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
3	2	2exbii 1876	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
4		19.42vv 1984	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
5		elvv 5737	. . . . . . 7 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
6	5	anbi2i 634	. . . . . 6 ⊢ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
7		vex 3467	. . . . . . 7 ⊢ 𝑧 ∈ V
8	7	biantru 538	. . . . . 6 ⊢ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V))
9	4, 6, 8	3bitr2i 302	. . . . 5 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V))
10		anass 473	. . . . 5 ⊢ (((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
11	3, 9, 10	3bitrri 301	. . . 4 ⊢ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
12	11	2exbii 1876	. . 3 ⊢ (∃𝑤∃𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑤∃𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
13		exrot4 2207	. . 3 ⊢ (∃𝑥∃𝑦∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑤∃𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
14		excom 2203	. . . . 5 ⊢ (∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
15		opex 5446	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
16		opeq1 4842	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
17	16	eqeq2d 2780	. . . . . . 7 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑤, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
18	15, 17	ceqsexv 3511	. . . . . 6 ⊢ (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
19	18	exbii 1875	. . . . 5 ⊢ (∃𝑧∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
20	14, 19	bitri 278	. . . 4 ⊢ (∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
21	20	2exbii 1876	. . 3 ⊢ (∃𝑥∃𝑦∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
22	12, 13, 21	3bitr2i 302	. 2 ⊢ (∃𝑤∃𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
23	1, 22	bitri 278	1 ⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   × cxp 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5178  df-xp 5668

This theorem is referenced by:  ssrelrel  5783  dftpos3  8240