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Theorem unisng 4484
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 (𝐴𝑉 {𝐴} = 𝐴)

Proof of Theorem unisng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 4220 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21unieqd 4478 . . 3 (𝑥 = 𝐴 {𝑥} = {𝐴})
3 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
42, 3eqeq12d 2666 . 2 (𝑥 = 𝐴 → ( {𝑥} = 𝑥 {𝐴} = 𝐴))
5 vex 3234 . . 3 𝑥 ∈ V
65unisn 4483 . 2 {𝑥} = 𝑥
74, 6vtoclg 3297 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {csn 4210   cuni 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213  df-uni 4469
This theorem is referenced by:  unisn3  4485  dfnfc2  4486  dfnfc2OLD  4487  unisn2  4827  en2other2  8870  pmtrprfv  17919  dprdsn  18481  indistopon  20853  ordtuni  21042  cmpcld  21253  ptcmplem5  21907  cldsubg  21961  icccmplem2  22673  vmappw  24887  chsupsn  28400  xrge0tsmseq  29915  esumsnf  30254  prsiga  30322  rossros  30371  cvmscld  31381  unisnif  32157  topjoin  32485  fnejoin2  32489  bj-snmoore  33193  heiborlem8  33747  fourierdlem80  40721
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