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Theorem unisng 3785
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 3592 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21unieqd 3779 . . 3  |-  ( x  =  A  ->  U. {
x }  =  U. { A } )
3 id 21 . . 3  |-  ( x  =  A  ->  x  =  A )
42, 3eqeq12d 2270 . 2  |-  ( x  =  A  ->  ( U. { x }  =  x 
<-> 
U. { A }  =  A ) )
5 vex 2743 . . 3  |-  x  e. 
_V
65unisn 3784 . 2  |-  U. {
x }  =  x
74, 6vtoclg 2794 1  |-  ( A  e.  V  ->  U. { A }  =  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   {csn 3581   U.cuni 3768
This theorem is referenced by:  dfnfc2  3786  unisn2  4459  unisn3  4460  dprdsn  15198  indistopon  16665  ordtuni  16847  cmpcld  17056  ptcmplem5  17677  cldsubg  17720  icccmplem2  18255  vmappw  20281  chsupsn  21917  cvmscld  23141  unisnif  23804  unexun  24901  topjoin  25646  fnejoin2  25650  heiborlem8  25874  en2other2  26714  pmtrprfv  26728
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 2237
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 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rex 2521  df-v 2742  df-un 3099  df-sn 3587  df-pr 3588  df-uni 3769
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