Theorem unisng 4452

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 4187	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	unieqd 4446	. . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥} = ∪ {𝐴})
3		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2637	. 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥} = 𝑥 ↔ ∪ {𝐴} = 𝐴))
5		vex 3203	. . 3 ⊢ 𝑥 ∈ V
6	5	unisn 4451	. 2 ⊢ ∪ {𝑥} = 𝑥
7	4, 6	vtoclg 3266	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {csn 4177  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  unisn3  4453  dfnfc2  4454  dfnfc2OLD  4455  unisn2  4794  en2other2  8832  pmtrprfv  17873  dprdsn  18435  indistopon  20805  ordtuni  20994  cmpcld  21205  ptcmplem5  21860  cldsubg  21914  icccmplem2  22626  vmappw  24842  chsupsn  28272  xrge0tsmseq  29787  esumsnf  30126  prsiga  30194  rossros  30243  cvmscld  31255  unisnif  32032  topjoin  32360  fnejoin2  32364  bj-snmoore  33068  heiborlem8  33617  fourierdlem80  40403