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Theorem sneqr 4167
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 4026 . . 3  |-  A  e. 
{ A }
3 eleq2 2496 . . 3  |-  ( { A }  =  { B }  ->  ( A  e.  { A }  <->  A  e.  { B }
) )
42, 3mpbii 214 . 2  |-  ( { A }  =  { B }  ->  A  e. 
{ B } )
51elsnc 4022 . 2  |-  ( A  e.  { B }  <->  A  =  B )
64, 5sylib 199 1  |-  ( { A }  =  { B }  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   _Vcvv 3080   {csn 3998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-sn 3999
This theorem is referenced by:  snsssn  4168  sneqrg  4169  opth1  4694  opthwiener  4722  canth2  7734  axcc2lem  8873  hashge3el3dif  12646  dis2ndc  20473  axlowdim1  24987  bj-snsetex  31525  poimirlem13  31917  poimirlem14  31918  wopprc  35855  hoidmv1le  38320  propeqop  38865  funsndifnop  38885
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