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Theorem sneqr 3754
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 3641 . . 3  |-  A  e. 
{ A }
3 eleq2 2319 . . 3  |-  ( { A }  =  { B }  ->  ( A  e.  { A }  <->  A  e.  { B }
) )
42, 3mpbii 204 . 2  |-  ( { A }  =  { B }  ->  A  e. 
{ B } )
51elsnc 3637 . 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 1619    e. wcel 1621   _Vcvv 2763   {csn 3614
This theorem is referenced by:  snsssn  3755  sneqrg  3756  opth1  4216  opthwiener  4240  canth2  6982  axcc2lem  8030  dis2ndc  17148  axlowdim1  23962  wopprc  26490
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-sn 3620
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