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Theorem elvvuni 5622
 Description: An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
elvvuni (𝐴 ∈ (V × V) → 𝐴𝐴)

Proof of Theorem elvvuni
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 5620 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3497 . . . . . 6 𝑥 ∈ V
3 vex 3497 . . . . . 6 𝑦 ∈ V
42, 3uniop 5397 . . . . 5 𝑥, 𝑦⟩ = {𝑥, 𝑦}
52, 3opi2 5353 . . . . 5 {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦
64, 5eqeltri 2909 . . . 4 𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦
7 unieq 4839 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
8 id 22 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩)
97, 8eleq12d 2907 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → ( 𝐴𝐴𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩))
106, 9mpbiri 260 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴𝐴)
1110exlimivv 1929 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴𝐴)
121, 11sylbi 219 1 (𝐴 ∈ (V × V) → 𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3494  {cpr 4562  ⟨cop 4566  ∪ cuni 4831   × cxp 5547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-opab 5121  df-xp 5555 This theorem is referenced by:  unielxp  7721
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