Theorem unisng 4925

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 4639	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	unieqi 4919	. . 3 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴}
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = ∪ {𝐴, 𝐴})
4		uniprg 4923	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
5	4	anidms 566	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
6		unidm 4157	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
7	6	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 𝐴) = 𝐴)
8	3, 5, 7	3eqtrd 2781	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ∪ cun 3949  {csn 4626  {cpr 4628  ∪ cuni 4907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908

This theorem is referenced by:  unisn  4926  unisn3  4928  dfnfc2  4929  unisn2  5312  unisucs  6461  en2other2  10049  pmtrprfv  19471  dprdsn  20056  indistopon  23008  ordtuni  23198  cmpcld  23410  ptcmplem5  24064  cldsubg  24119  icccmplem2  24845  vmappw  27159  chsupsn  31432  xrge0tsmseq  33067  cycpm2tr  33139  qustrivr  33393  esumsnf  34065  prsiga  34132  rossros  34181  cvmscld  35278  unisnif  35926  topjoin  36366  fnejoin2  36370  bj-snmoore  37114  pibt2  37418  heiborlem8  37825  sucunisn  43384  onsucunitp  43386  oaun3  43395  fourierdlem80  46201