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Theorem sneqr 4199
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 4060 . . 3  |-  A  e. 
{ A }
3 eleq2 2530 . . 3  |-  ( { A }  =  { B }  ->  ( A  e.  { A }  <->  A  e.  { B }
) )
42, 3mpbii 211 . 2  |-  ( { A }  =  { B }  ->  A  e. 
{ B } )
51elsnc 4056 . 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 1395    e. wcel 1819   _Vcvv 3109   {csn 4032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sn 4033
This theorem is referenced by:  snsssn  4200  sneqrg  4201  opth1  4729  opthwiener  4758  canth2  7689  axcc2lem  8833  hashge3el3dif  12527  dis2ndc  20086  axlowdim1  24388  wopprc  31134  bj-snsetex  34622
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