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Theorem sneqr 4152
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 4008 . . 3  |-  A  e. 
{ A }
3 eleq2 2529 . . 3  |-  ( { A }  =  { B }  ->  ( A  e.  { A }  <->  A  e.  { B }
) )
42, 3mpbii 216 . 2  |-  ( { A }  =  { B }  ->  A  e. 
{ B } )
51elsnc 4004 . 2  |-  ( A  e.  { B }  <->  A  =  B )
64, 5sylib 201 1  |-  ( { A }  =  { B }  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455    e. wcel 1898   _Vcvv 3057   {csn 3980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sn 3981
This theorem is referenced by:  snsssn  4153  sneqrg  4154  opth1  4692  opthwiener  4720  canth2  7756  axcc2lem  8897  hashge3el3dif  12681  dis2ndc  20530  axlowdim1  25045  bj-snsetex  31603  poimirlem13  31999  poimirlem14  32000  wopprc  35931  hoidmv1le  38523  propeqop  39136  funsndifnop  39160
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