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Theorem sneqbg 3778
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 3777 . 2  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
2 sneq 3618 . 2  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2impbid1 142 1  |-  ( A  e.  V  ->  ( { A }  =  { B }  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2160   {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sn 3613
This theorem is referenced by:  infpwfidom  7228
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