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Theorem elvvuni 5319
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 5317 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3354 . . . . . 6 𝑥 ∈ V
3 vex 3354 . . . . . 6 𝑦 ∈ V
42, 3uniop 5108 . . . . 5 𝑥, 𝑦⟩ = {𝑥, 𝑦}
52, 3opi2 5065 . . . . 5 {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦
64, 5eqeltri 2846 . . . 4 𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦
7 unieq 4582 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
8 id 22 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩)
97, 8eleq12d 2844 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → ( 𝐴𝐴𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩))
106, 9mpbiri 248 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴𝐴)
1110exlimivv 2012 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴𝐴)
121, 11sylbi 207 1 (𝐴 ∈ (V × V) → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  {cpr 4318  cop 4322   cuni 4574   × cxp 5247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-opab 4847  df-xp 5255
This theorem is referenced by:  unielxp  7353
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