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Theorem elsncg 3828
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
elsncg  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )

Proof of Theorem elsncg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2441 . 2  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 df-sn 3812 . 2  |-  { B }  =  { x  |  x  =  B }
31, 2elab2g 3076 1  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {csn 3806
This theorem is referenced by:  elsnc  3829  elsni  3830  snidg  3831  eltpg  3843  eldifsn  3919  elsucg  4640  ltxr  10705  elfzp12  11116  ramcl  13387  nbcusgra  21462  xrge0tsmsbi  24214  elzrhunit  24353  elzdif0  24354  frgrancvvdeqlem1  28320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sn 3812
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