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Theorem unisng 4418
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 4158 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21unieqd 4412 . . 3 (𝑥 = 𝐴 {𝑥} = {𝐴})
3 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
42, 3eqeq12d 2636 . 2 (𝑥 = 𝐴 → ( {𝑥} = 𝑥 {𝐴} = 𝐴))
5 vex 3189 . . 3 𝑥 ∈ V
65unisn 4417 . 2 {𝑥} = 𝑥
74, 6vtoclg 3252 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {csn 4148   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3188  df-un 3560  df-sn 4149  df-pr 4151  df-uni 4403
This theorem is referenced by:  unisn3  4419  dfnfc2  4420  dfnfc2OLD  4421  unisn2  4754  en2other2  8776  pmtrprfv  17794  dprdsn  18356  indistopon  20715  ordtuni  20904  cmpcld  21115  ptcmplem5  21770  cldsubg  21824  icccmplem2  22534  vmappw  24742  chsupsn  28118  xrge0tsmseq  29569  esumsnf  29904  prsiga  29972  rossros  30021  cvmscld  30960  unisnif  31671  topjoin  31999  fnejoin2  32003  heiborlem8  33246  fourierdlem80  39707
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