Theorem elvvuni 4788

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 4786	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2803	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2803	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	uniop 4346	. . . . 5 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
5	2, 3	opi2 4323	. . . . 5 ⊢ {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦⟩
6	4, 5	eqeltri 2302	. . . 4 ⊢ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩
7		unieq 3900	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩)
8		id 19	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩)
9	7, 8	eleq12d 2300	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (∪ 𝐴 ∈ 𝐴 ↔ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩))
10	6, 9	mpbiri 168	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 ∈ 𝐴)
11	10	exlimivv 1943	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 ∈ 𝐴)
12	1, 11	sylbi 121	1 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800  {cpr 3668  ⟨cop 3670  ∪ cuni 3891   × cxp 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-opab 4149  df-xp 4729

This theorem is referenced by:  unielxp  6332