MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elvvuni Structured version   Visualization version   GIF version

Theorem elvvuni 5663
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 5661 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3436 . . . . . 6 𝑥 ∈ V
3 vex 3436 . . . . . 6 𝑦 ∈ V
42, 3uniop 5429 . . . . 5 𝑥, 𝑦⟩ = {𝑥, 𝑦}
52, 3opi2 5384 . . . . 5 {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦
64, 5eqeltri 2835 . . . 4 𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦
7 unieq 4850 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
8 id 22 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩)
97, 8eleq12d 2833 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → ( 𝐴𝐴𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩))
106, 9mpbiri 257 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴𝐴)
1110exlimivv 1935 . 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 1539  wex 1782  wcel 2106  Vcvv 3432  {cpr 4563  cop 4567   cuni 4839   × cxp 5587
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-opab 5137  df-xp 5595
This theorem is referenced by:  unielxp  7869
  Copyright terms: Public domain W3C validator