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Theorem elvvuni 5713
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 5711 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3452 . . . . . 6 𝑥 ∈ V
3 vex 3452 . . . . . 6 𝑦 ∈ V
42, 3uniop 5477 . . . . 5 𝑥, 𝑦⟩ = {𝑥, 𝑦}
52, 3opi2 5431 . . . . 5 {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦
64, 5eqeltri 2834 . . . 4 𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦
7 unieq 4881 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
8 id 22 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩)
97, 8eleq12d 2832 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → ( 𝐴𝐴𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩))
106, 9mpbiri 258 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴𝐴)
1110exlimivv 1936 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴𝐴)
121, 11sylbi 216 1 (𝐴 ∈ (V × V) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wex 1782  wcel 2107  Vcvv 3448  {cpr 4593  cop 4597   cuni 4870   × cxp 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-opab 5173  df-xp 5644
This theorem is referenced by:  unielxp  7964
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