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Theorem elvvv 5149
Description: Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
Assertion
Ref Expression
elvvv (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem elvvv
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elxp 5101 . 2 (𝐴 ∈ ((V × V) × V) ↔ ∃𝑤𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
2 ancom 466 . . . . . 6 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
322exbii 1772 . . . . 5 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
4 19.42vv 1917 . . . . . 6 (∃𝑥𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
5 elvv 5148 . . . . . . 7 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
65anbi2i 729 . . . . . 6 ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
7 vex 3193 . . . . . . 7 𝑧 ∈ V
87biantru 526 . . . . . 6 ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V))
94, 6, 83bitr2i 288 . . . . 5 (∃𝑥𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V))
10 anass 680 . . . . 5 (((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
113, 9, 103bitrri 287 . . . 4 ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
12112exbii 1772 . . 3 (∃𝑤𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑤𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
13 exrot4 2043 . . 3 (∃𝑥𝑦𝑤𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑤𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
14 excom 2039 . . . . 5 (∃𝑤𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
15 opex 4903 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
16 opeq1 4377 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
1716eqeq2d 2631 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑤, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
1815, 17ceqsexv 3232 . . . . . 6 (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
1918exbii 1771 . . . . 5 (∃𝑧𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
2014, 19bitri 264 . . . 4 (∃𝑤𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
21202exbii 1772 . . 3 (∃𝑥𝑦𝑤𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
2212, 13, 213bitr2i 288 . 2 (∃𝑤𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
231, 22bitri 264 1 (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3190  cop 4161   × cxp 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-opab 4684  df-xp 5090
This theorem is referenced by:  ssrelrel  5191  dftpos3  7330
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