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Theorem snidb 3998
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 3997 . 2  |-  ( A  e.  _V  ->  A  e.  { A } )
2 elex 3067 . 2  |-  ( A  e.  { A }  ->  A  e.  _V )
31, 2impbii 188 1  |-  ( A  e.  _V  <->  A  e.  { A } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1842   _Vcvv 3058   {csn 3971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-sn 3972
This theorem is referenced by:  snid  3999  dffv2  5921
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