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Theorem unisng 3818
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
StepHypRef Expression
1 sneq 3625 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21unieqd 3812 . . 3  |-  ( x  =  A  ->  U. {
x }  =  U. { A } )
3 id 21 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2272 . 2  |-  ( x  =  A  ->  ( U. { x }  =  x 
<-> 
U. { A }  =  A ) )
5 vex 2766 . . 3  |-  x  e. 
_V
65unisn 3817 . 2  |-  U. {
x }  =  x
74, 6vtoclg 2818 1  |-  ( A  e.  V  ->  U. { A }  =  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   {csn 3614   U.cuni 3801
This theorem is referenced by:  dfnfc2  3819  unisn2  4494  unisn3  4495  dprdsn  15234  indistopon  16701  ordtuni  16883  cmpcld  17092  ptcmplem5  17713  cldsubg  17756  icccmplem2  18291  vmappw  20317  chsupsn  21953  cvmscld  23177  unisnif  23840  unexun  24937  topjoin  25682  fnejoin2  25686  heiborlem8  25910  en2other2  26750  pmtrprfv  26764
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rex 2524  df-v 2765  df-un 3132  df-sn 3620  df-pr 3621  df-uni 3802
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