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Theorem sneqr 3910
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 3786 . . 3  |-  A  e. 
{ A }
3 eleq2 2450 . . 3  |-  ( { A }  =  { B }  ->  ( A  e.  { A }  <->  A  e.  { B }
) )
42, 3mpbii 203 . 2  |-  ( { A }  =  { B }  ->  A  e. 
{ B } )
51elsnc 3782 . 2  |-  ( A  e.  { B }  <->  A  =  B )
64, 5sylib 189 1  |-  ( { A }  =  { B }  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2901   {csn 3759
This theorem is referenced by:  snsssn  3911  sneqrg  3912  opth1  4377  opthwiener  4401  canth2  7198  axcc2lem  8251  dis2ndc  17446  axlowdim1  25614  wopprc  26794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-sn 3765
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