Theorem unisng 4883

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 4595	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	unieqi 4877	. . 3 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴}
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = ∪ {𝐴, 𝐴})
4		uniprg 4881	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
5	4	anidms 566	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
6		unidm 4111	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
7	6	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 𝐴) = 𝐴)
8	3, 5, 7	3eqtrd 2776	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∪ cun 3901  {csn 4582  {cpr 4584  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-ss 3920  df-sn 4583  df-pr 4585  df-uni 4866

This theorem is referenced by:  unisn  4884  unisn3  4886  dfnfc2  4887  unisn2  5259  unisucs  6404  en2other2  9931  pmtrprfv  19394  dprdsn  19979  indistopon  22957  ordtuni  23146  cmpcld  23358  ptcmplem5  24012  cldsubg  24067  icccmplem2  24780  vmappw  27094  chsupsn  31500  xrge0tsmseq  33168  cycpm2tr  33212  qustrivr  33457  esumsnf  34241  prsiga  34308  rossros  34357  cvmscld  35486  unisnif  36136  topjoin  36578  fnejoin2  36582  bj-snmoore  37360  pibt2  37666  heiborlem8  38063  sucunisn  43722  onsucunitp  43724  oaun3  43733  fourierdlem80  46538