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Theorem snidb 3842
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 3841 . 2  |-  ( A  e.  _V  ->  A  e.  { A } )
2 elex 2966 . 2  |-  ( A  e.  { A }  ->  A  e.  _V )
31, 2impbii 182 1  |-  ( A  e.  _V  <->  A  e.  { A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1726   _Vcvv 2958   {csn 3816
This theorem is referenced by:  snid  3843  dffv2  5798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-v 2960  df-sn 3822
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