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Theorem sneqr 3930
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 3805 . . 3  |-  A  e. 
{ A }
3 eleq2 2469 . . 3  |-  ( { A }  =  { B }  ->  ( A  e.  { A }  <->  A  e.  { B }
) )
42, 3mpbii 203 . 2  |-  ( { A }  =  { B }  ->  A  e. 
{ B } )
51elsnc 3801 . 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 1721   _Vcvv 2920   {csn 3778
This theorem is referenced by:  snsssn  3931  sneqrg  3932  opth1  4398  opthwiener  4422  canth2  7223  axcc2lem  8276  dis2ndc  17480  axlowdim1  25806  wopprc  26995
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-sn 3784
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