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Theorem unisng 4034
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 3827 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21unieqd 4028 . . 3  |-  ( x  =  A  ->  U. {
x }  =  U. { A } )
3 id 21 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2452 . 2  |-  ( x  =  A  ->  ( U. { x }  =  x 
<-> 
U. { A }  =  A ) )
5 vex 2961 . . 3  |-  x  e. 
_V
65unisn 4033 . 2  |-  U. {
x }  =  x
74, 6vtoclg 3013 1  |-  ( A  e.  V  ->  U. { A }  =  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {csn 3816   U.cuni 4017
This theorem is referenced by:  dfnfc2  4035  unisn2  4714  unisn3  4715  dprdsn  15599  indistopon  17070  ordtuni  17259  cmpcld  17470  ptcmplem5  18092  cldsubg  18145  icccmplem2  18859  vmappw  20904  chsupsn  22920  xrge0tsmseq  24230  esumsn  24461  prsiga  24519  cvmscld  24965  unisnif  25775  topjoin  26408  fnejoin2  26412  heiborlem8  26541  en2other2  27373  pmtrprfv  27387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-v 2960  df-un 3327  df-sn 3822  df-pr 3823  df-uni 4018
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