Theorem elvvuni 4541

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 4539	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2644	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2644	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	uniop 4115	. . . . 5 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
5	2, 3	opi2 4093	. . . . 5 ⊢ {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦⟩
6	4, 5	eqeltri 2172	. . . 4 ⊢ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩
7		unieq 3692	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 = ∪ ⟨𝑥, 𝑦⟩)
8		id 19	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩)
9	7, 8	eleq12d 2170	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (∪ 𝐴 ∈ 𝐴 ↔ ∪ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩))
10	6, 9	mpbiri 167	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 ∈ 𝐴)
11	10	exlimivv 1835	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → ∪ 𝐴 ∈ 𝐴)
12	1, 11	sylbi 120	1 ⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1299  ∃wex 1436   ∈ wcel 1448  Vcvv 2641  {cpr 3475  ⟨cop 3477  ∪ cuni 3683   × cxp 4475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-opab 3930  df-xp 4483

This theorem is referenced by:  unielxp  6002