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Theorem sneqr 4052
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 3917 . . 3  |-  A  e. 
{ A }
3 eleq2 2504 . . 3  |-  ( { A }  =  { B }  ->  ( A  e.  { A }  <->  A  e.  { B }
) )
42, 3mpbii 211 . 2  |-  ( { A }  =  { B }  ->  A  e. 
{ B } )
51elsnc 3913 . 2  |-  ( A  e.  { B }  <->  A  =  B )
64, 5sylib 196 1  |-  ( { A }  =  { B }  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2984   {csn 3889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2986  df-sn 3890
This theorem is referenced by:  snsssn  4053  sneqrg  4054  opth1  4577  opthwiener  4605  canth2  7476  axcc2lem  8617  hashge3el3dif  12199  dis2ndc  19076  axlowdim1  23217  wopprc  29391  bj-snsetex  32468
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