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Theorem sneqbg 4150
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 4149 . 2  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
2 sneq 3994 . 2  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2impbid1 203 1  |-  ( A  e.  V  ->  ( { A }  =  { B }  <->  A  =  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   {csn 3984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3078  df-sn 3985
This theorem is referenced by:  suppval1  6805  suppsnop  6813  fseqdom  8306  infpwfidom  8308  canthwe  8928  s111  12419  altopthg  28141  altopthbg  28142  mat1dimelbas  31032  mat1dimbas  31033  bj-snglc  32779
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