Theorem elvvv 4648

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 4602	. 2 ⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑤∃𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
2		anass 399	. . . . 5 ⊢ (((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
3		19.42vv 1891	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
4		ancom 264	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
5	4	2exbii 1586	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
6		vex 2715	. . . . . . . 8 ⊢ 𝑧 ∈ V
7	6	biantru 300	. . . . . . 7 ⊢ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V))
8		elvv 4647	. . . . . . . 8 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
9	8	anbi2i 453	. . . . . . 7 ⊢ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
10	7, 9	bitr3i 185	. . . . . 6 ⊢ (((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
11	3, 5, 10	3bitr4ri 212	. . . . 5 ⊢ (((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
12	2, 11	bitr3i 185	. . . 4 ⊢ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
13	12	2exbii 1586	. . 3 ⊢ (∃𝑤∃𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑤∃𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
14		exrot4 1671	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑤∃𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
15		excom 1644	. . . . . 6 ⊢ (∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
16		vex 2715	. . . . . . . . 9 ⊢ 𝑥 ∈ V
17		vex 2715	. . . . . . . . 9 ⊢ 𝑦 ∈ V
18	16, 17	opex 4189	. . . . . . . 8 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
19		opeq1 3741	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
20	19	eqeq2d 2169	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑤, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
21	18, 20	ceqsexv 2751	. . . . . . 7 ⊢ (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
22	21	exbii 1585	. . . . . 6 ⊢ (∃𝑧∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
23	15, 22	bitri 183	. . . . 5 ⊢ (∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
24	23	2exbii 1586	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑤∃𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
25	14, 24	bitr3i 185	. . 3 ⊢ (∃𝑤∃𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
26	13, 25	bitri 183	. 2 ⊢ (∃𝑤∃𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
27	1, 26	bitri 183	1 ⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1335  ∃wex 1472   ∈ wcel 2128  Vcvv 2712  ⟨cop 3563   × cxp 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-xp 4591

This theorem is referenced by:  ssrelrel  4685  dftpos3  6206