MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unisng Structured version   Visualization version   GIF version

Theorem unisng 4930
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
StepHypRef Expression
1 dfsn2 4642 . . . 4 {𝐴} = {𝐴, 𝐴}
21unieqi 4922 . . 3 {𝐴} = {𝐴, 𝐴}
32a1i 11 . 2 (𝐴𝑉 {𝐴} = {𝐴, 𝐴})
4 uniprg 4926 . . 3 ((𝐴𝑉𝐴𝑉) → {𝐴, 𝐴} = (𝐴𝐴))
54anidms 568 . 2 (𝐴𝑉 {𝐴, 𝐴} = (𝐴𝐴))
6 unidm 4153 . . 3 (𝐴𝐴) = 𝐴
76a1i 11 . 2 (𝐴𝑉 → (𝐴𝐴) = 𝐴)
83, 5, 73eqtrd 2777 1 (𝐴𝑉 {𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cun 3947  {csn 4629  {cpr 4631   cuni 4909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632  df-uni 4910
This theorem is referenced by:  unisn  4931  unisn3  4933  dfnfc2  4934  unisn2  5313  unisucs  6442  en2other2  10004  pmtrprfv  19321  dprdsn  19906  indistopon  22504  ordtuni  22694  cmpcld  22906  ptcmplem5  23560  cldsubg  23615  icccmplem2  24339  vmappw  26620  chsupsn  30666  xrge0tsmseq  32211  cycpm2tr  32278  qustrivr  32477  esumsnf  33062  prsiga  33129  rossros  33178  cvmscld  34264  unisnif  34897  topjoin  35250  fnejoin2  35254  bj-snmoore  35994  pibt2  36298  heiborlem8  36686  sucunisn  42121  onsucunitp  42123  oaun3  42132  fourierdlem80  44902
  Copyright terms: Public domain W3C validator