Theorem unisng 4388

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

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4135	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	unieqd 4382	. . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥} = ∪ {𝐴})
3		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2625	. 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥} = 𝑥 ↔ ∪ {𝐴} = 𝐴))
5		vex 3176	. . 3 ⊢ 𝑥 ∈ V
6	5	unisn 4387	. 2 ⊢ ∪ {𝑥} = 𝑥
7	4, 6	vtoclg 3239	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {csn 4125  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373

This theorem is referenced by:  unisn3  4389  dfnfc2  4390  dfnfc2OLD  4391  unisn2  4722  en2other2  8715  pmtrprfv  17696  dprdsn  18258  indistopon  20615  ordtuni  20804  cmpcld  21015  ptcmplem5  21670  cldsubg  21724  icccmplem2  22434  vmappw  24642  chsupsn  27656  xrge0tsmseq  29118  esumsnf  29453  prsiga  29521  rossros  29570  cvmscld  30509  unisnif  31202  topjoin  31530  fnejoin2  31534  heiborlem8  32787  fourierdlem80  39079