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Theorem snidb 3666
Description: A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
snidb  |-  ( A  e.  _V  <->  A  e.  { A } )

Proof of Theorem snidb
StepHypRef Expression
1 snidg 3665 . 2  |-  ( A  e.  _V  ->  A  e.  { A } )
2 elex 2796 . 2  |-  ( A  e.  { A }  ->  A  e.  _V )
31, 2impbii 180 1  |-  ( A  e.  _V  <->  A  e.  { A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   _Vcvv 2788   {csn 3640
This theorem is referenced by:  snid  3667  dffv2  5592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sn 3646
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