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Theorem elvvv 4570
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 4524 . 2 (𝐴 ∈ ((V × V) × V) ↔ ∃𝑤𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
2 anass 396 . . . . 5 (((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)))
3 19.42vv 1863 . . . . . 6 (∃𝑥𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
4 ancom 264 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
542exbii 1568 . . . . . 6 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
6 vex 2661 . . . . . . . 8 𝑧 ∈ V
76biantru 298 . . . . . . 7 ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V))
8 elvv 4569 . . . . . . . 8 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
98anbi2i 450 . . . . . . 7 ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
107, 9bitr3i 185 . . . . . 6 (((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V) ↔ (𝐴 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
113, 5, 103bitr4ri 212 . . . . 5 (((𝐴 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 ∈ (V × V)) ∧ 𝑧 ∈ V) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
122, 11bitr3i 185 . . . 4 ((𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
13122exbii 1568 . . 3 (∃𝑤𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑤𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
14 exrot4 1652 . . . 4 (∃𝑥𝑦𝑤𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑤𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
15 excom 1625 . . . . . 6 (∃𝑤𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩))
16 vex 2661 . . . . . . . . 9 𝑥 ∈ V
17 vex 2661 . . . . . . . . 9 𝑦 ∈ V
1816, 17opex 4119 . . . . . . . 8 𝑥, 𝑦⟩ ∈ V
19 opeq1 3673 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
2019eqeq2d 2127 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑤, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
2118, 20ceqsexv 2697 . . . . . . 7 (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
2221exbii 1567 . . . . . 6 (∃𝑧𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
2315, 22bitri 183 . . . . 5 (∃𝑤𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
24232exbii 1568 . . . 4 (∃𝑥𝑦𝑤𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
2514, 24bitr3i 185 . . 3 (∃𝑤𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝐴 = ⟨𝑤, 𝑧⟩) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
2613, 25bitri 183 . 2 (∃𝑤𝑧(𝐴 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 ∈ (V × V) ∧ 𝑧 ∈ V)) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
271, 26bitri 183 1 (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  cop 3498   × cxp 4505
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-xp 4513
This theorem is referenced by:  ssrelrel  4607  dftpos3  6125
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