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Theorem snidb 4049
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 4048 . 2  |-  ( A  e.  _V  ->  A  e.  { A } )
2 elex 3117 . 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 1762   _Vcvv 3108   {csn 4022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-v 3110  df-sn 4023
This theorem is referenced by:  snid  4050  dffv2  5933
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