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Theorem sneqbg 3660
Description: Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
sneqbg  |-  ( A  e.  V  ->  ( { A }  =  { B }  <->  A  =  B
) )

Proof of Theorem sneqbg
StepHypRef Expression
1 sneqrg 3659 . 2  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
2 sneq 3508 . 2  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2impbid1 141 1  |-  ( A  e.  V  ->  ( { A }  =  { B }  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465   {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sn 3503
This theorem is referenced by:  infpwfidom  7022
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