Theorem unisng 4882

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 4594	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	unieqi 4876	. . 3 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴}
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = ∪ {𝐴, 𝐴})
4		uniprg 4880	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
5	4	anidms 574	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
6		unidm 4110	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
7	6	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 𝐴) = 𝐴)
8	3, 5, 7	3eqtrd 2800	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141   ∪ cun 3902  {csn 4581  {cpr 4583  ∪ cuni 4864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-ss 3921  df-sn 4582  df-pr 4584  df-uni 4865

This theorem is referenced by:  unisn  4883  unisn3  4885  dfnfc2  4886  unisn2  5261  unisucs  6421  en2other2  9962  pmtrprfv  19476  dprdsn  20061  indistopon  23041  ordtuni  23230  cmpcld  23442  ptcmplem5  24096  cldsubg  24151  icccmplem2  24864  vmappw  27157  chsupsn  31562  xrge0tsmseq  33216  cycpm2tr  33260  qustrivr  33512  esumsnf  34322  prsiga  34389  rossros  34438  cvmscld  35587  unisnif  36237  topjoin  36689  fnejoin2  36693  bj-snmoore  37567  pibt2  37875  heiborlem8  38281  sucunisn  43912  onsucunitp  43914  oaun3  43923  fourierdlem80  46724