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Theorem sneqr 3781
Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)
Hypothesis
Ref Expression
sneqr.1  |-  A  e. 
_V
Assertion
Ref Expression
sneqr  |-  ( { A }  =  { B }  ->  A  =  B )

Proof of Theorem sneqr
StepHypRef Expression
1 sneqr.1 . . . 4  |-  A  e. 
_V
21snid 3668 . . 3  |-  A  e. 
{ A }
3 eleq2 2345 . . 3  |-  ( { A }  =  { B }  ->  ( A  e.  { A }  <->  A  e.  { B }
) )
42, 3mpbii 204 . 2  |-  ( { A }  =  { B }  ->  A  e. 
{ B } )
51elsnc 3664 . 2  |-  ( A  e.  { B }  <->  A  =  B )
64, 5sylib 190 1  |-  ( { A }  =  { B }  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   _Vcvv 2789   {csn 3641
This theorem is referenced by:  snsssn  3782  sneqrg  3783  opth1  4243  opthwiener  4267  canth2  7009  axcc2lem  8057  dis2ndc  17180  axlowdim1  23994  wopprc  26522
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-sn 3647
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