Theorem elvvuni 4331

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 4329	. 2 ⊢ (A ∈ (V × V) ↔ ∃x∃y A = ⟨x, y⟩)
2		vex 2538	. . . . . 6 ⊢ x ∈ V
3		vex 2538	. . . . . 6 ⊢ y ∈ V
4	2, 3	uniop 3966	. . . . 5 ⊢ ∪ ⟨x, y⟩ = {x, y}
5	2, 3	opi2 3944	. . . . 5 ⊢ {x, y} ∈ ⟨x, y⟩
6	4, 5	eqeltri 2092	. . . 4 ⊢ ∪ ⟨x, y⟩ ∈ ⟨x, y⟩
7		unieq 3563	. . . . 5 ⊢ (A = ⟨x, y⟩ → ∪ A = ∪ ⟨x, y⟩)
8		id 19	. . . . 5 ⊢ (A = ⟨x, y⟩ → A = ⟨x, y⟩)
9	7, 8	eleq12d 2090	. . . 4 ⊢ (A = ⟨x, y⟩ → (∪ A ∈ A ↔ ∪ ⟨x, y⟩ ∈ ⟨x, y⟩))
10	6, 9	mpbiri 157	. . 3 ⊢ (A = ⟨x, y⟩ → ∪ A ∈ A)
11	10	exlimivv 1758	. 2 ⊢ (∃x∃y A = ⟨x, y⟩ → ∪ A ∈ A)
12	1, 11	sylbi 114	1 ⊢ (A ∈ (V × V) → ∪ A ∈ A)

Syntax hints:   → wi 4   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535  {cpr 3351  ⟨cop 3353  ∪ cuni 3554   × cxp 4270

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-opab 3793  df-xp 4278

This theorem is referenced by:  unielxp  5723