Theorem unisng 4819

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 4538	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	unieqi 4813	. . 3 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴}
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = ∪ {𝐴, 𝐴})
4		uniprg 4818	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
5	4	anidms 570	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴))
6		unidm 4079	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
7	6	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 𝐴) = 𝐴)
8	3, 5, 7	3eqtrd 2837	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ∪ cun 3879  {csn 4525  {cpr 4527  ∪ cuni 4800

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-uni 4801

This theorem is referenced by:  unisn  4820  unisn3  4821  dfnfc2  4822  unisn2  5180  en2other2  9420  pmtrprfv  18573  dprdsn  19151  indistopon  21606  ordtuni  21795  cmpcld  22007  ptcmplem5  22661  cldsubg  22716  icccmplem2  23428  vmappw  25701  chsupsn  29196  xrge0tsmseq  30744  cycpm2tr  30811  qustrivr  30981  esumsnf  31433  prsiga  31500  rossros  31549  cvmscld  32633  unisnif  33499  topjoin  33826  fnejoin2  33830  bj-snmoore  34528  pibt2  34834  heiborlem8  35256  fourierdlem80  42828