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Theorem sneqbg 3883
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 3882 . 2  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
2 sneq 3740 . 2  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2impbid1 194 1  |-  ( A  e.  V  ->  ( { A }  =  { B }  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1647    e. wcel 1715   {csn 3729
This theorem is referenced by:  fseqdom  7800  infpwfidom  7802  canthwe  8420  s111  11649  altopthg  25243  altopthbg  25244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-sn 3735
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