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

Theorem unisng 3992
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  e.  V  ->  U. { A }  =  A
)

Proof of Theorem unisng
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sneq 3785 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21unieqd 3986 . . 3  |-  ( x  =  A  ->  U. {
x }  =  U. { A } )
3 id 20 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2418 . 2  |-  ( x  =  A  ->  ( U. { x }  =  x 
<-> 
U. { A }  =  A ) )
5 vex 2919 . . 3  |-  x  e. 
_V
65unisn 3991 . 2  |-  U. {
x }  =  x
74, 6vtoclg 2971 1  |-  ( A  e.  V  ->  U. { A }  =  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {csn 3774   U.cuni 3975
This theorem is referenced by:  dfnfc2  3993  unisn2  4670  unisn3  4671  dprdsn  15549  indistopon  17020  ordtuni  17208  cmpcld  17419  ptcmplem5  18040  cldsubg  18093  icccmplem2  18807  vmappw  20852  chsupsn  22868  xrge0tsmseq  24178  esumsn  24409  prsiga  24467  cvmscld  24913  unisnif  25678  topjoin  26284  fnejoin2  26288  heiborlem8  26417  en2other2  27250  pmtrprfv  27264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-v 2918  df-un 3285  df-sn 3780  df-pr 3781  df-uni 3976
  Copyright terms: Public domain W3C validator