Theorem unisng 4929

Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)

Assertion

Ref	Expression
unisng	⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Proof of Theorem unisng

Step	Hyp	Ref	Expression
1		dfsn2 4641	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	unieqi 4921	. . 3 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴}
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = ∪ {𝐴, 𝐴})
4		uniprg 4925	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
5	4	anidms 566	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
6		unidm 4152	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
7	6	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 𝐴) = 𝐴)
8	3, 5, 7	3eqtrd 2775	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105   ∪ cun 3946  {csn 4628  {cpr 4630  ∪ cuni 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-uni 4909

This theorem is referenced by:  unisn  4930  unisn3  4932  dfnfc2  4933  unisn2  5312  unisucs  6441  en2other2  10010  pmtrprfv  19369  dprdsn  19954  indistopon  22824  ordtuni  23014  cmpcld  23226  ptcmplem5  23880  cldsubg  23935  icccmplem2  24659  vmappw  26961  chsupsn  31099  xrge0tsmseq  32647  cycpm2tr  32714  qustrivr  32917  esumsnf  33526  prsiga  33593  rossros  33642  cvmscld  34728  unisnif  35367  topjoin  35714  fnejoin2  35718  bj-snmoore  36458  pibt2  36762  heiborlem8  37150  sucunisn  42584  onsucunitp  42586  oaun3  42595  fourierdlem80  45361