Theorem elvvuni 4347

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 4345	. 2 ⊢ (A ∈ (V × V) ↔ ∃x∃y A = ⟨x, y⟩)
2		vex 2554	. . . . . 6 ⊢ x ∈ V
3		vex 2554	. . . . . 6 ⊢ y ∈ V
4	2, 3	uniop 3983	. . . . 5 ⊢ ∪ ⟨x, y⟩ = {x, y}
5	2, 3	opi2 3961	. . . . 5 ⊢ {x, y} ∈ ⟨x, y⟩
6	4, 5	eqeltri 2107	. . . 4 ⊢ ∪ ⟨x, y⟩ ∈ ⟨x, y⟩
7		unieq 3580	. . . . 5 ⊢ (A = ⟨x, y⟩ → ∪ A = ∪ ⟨x, y⟩)
8		id 19	. . . . 5 ⊢ (A = ⟨x, y⟩ → A = ⟨x, y⟩)
9	7, 8	eleq12d 2105	. . . 4 ⊢ (A = ⟨x, y⟩ → (∪ A ∈ A ↔ ∪ ⟨x, y⟩ ∈ ⟨x, y⟩))
10	6, 9	mpbiri 157	. . 3 ⊢ (A = ⟨x, y⟩ → ∪ A ∈ A)
11	10	exlimivv 1773	. 2 ⊢ (∃x∃y A = ⟨x, y⟩ → ∪ A ∈ A)
12	1, 11	sylbi 114	1 ⊢ (A ∈ (V × V) → ∪ A ∈ A)

Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  {cpr 3368  ⟨cop 3370  ∪ cuni 3571   × cxp 4286

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-opab 3810  df-xp 4294

This theorem is referenced by:  unielxp  5742