Theorem elvvuni 5736

Description: An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)

Assertion

Ref	Expression
elvvuni	⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)

Proof of Theorem elvvuni

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elvv 5734	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3468	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3468	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	uniop 5495	. . . . 5 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
5	2, 3	opi2 5449	. . . . 5 ⊢ {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦⟩
6	4, 5	eqeltri 2831	. . . 4 ⊢ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩
7		unieq 4899	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩)
8		id 22	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩)
9	7, 8	eleq12d 2829	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (∪ 𝐴 ∈ 𝐴 ↔ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩))
10	6, 9	mpbiri 258	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 ∈ 𝐴)
11	10	exlimivv 1932	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 ∈ 𝐴)
12	1, 11	sylbi 217	1 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3464  {cpr 4608  ⟨cop 4612  ∪ cuni 4888   × cxp 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-opab 5187  df-xp 5665

This theorem is referenced by:  unielxp  8031