Theorem elvvuni 5776

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 5774	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3492	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3492	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	uniop 5534	. . . . 5 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
5	2, 3	opi2 5489	. . . . 5 ⊢ {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦⟩
6	4, 5	eqeltri 2840	. . . 4 ⊢ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩
7		unieq 4942	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩)
8		id 22	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩)
9	7, 8	eleq12d 2838	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (∪ 𝐴 ∈ 𝐴 ↔ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩))
10	6, 9	mpbiri 258	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 ∈ 𝐴)
11	10	exlimivv 1931	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 ∈ 𝐴)
12	1, 11	sylbi 217	1 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488  {cpr 4650  ⟨cop 4654  ∪ cuni 4931   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-opab 5229  df-xp 5706

This theorem is referenced by:  unielxp  8068