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Theorem sneqbg 3783
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 3782 . 2  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
2 sneq 3651 . 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 1623    e. wcel 1684   {csn 3640
This theorem is referenced by:  fseqdom  7653  infpwfidom  7655  canthwe  8273  s111  11448  altopthg  24501  altopthbg  24502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sn 3646
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