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Theorem elsncg 3664
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 2291 . 2  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 df-sn 3648 . 2  |-  { B }  =  { x  |  x  =  B }
31, 2elab2g 2918 1  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1625    e. wcel 1686   {csn 3642
This theorem is referenced by:  elsnc  3665  elsni  3666  snidg  3667  eltpg  3678  eldifsn  3751  elsucg  4461  ltxr  10459  elfzp12  10863  ramcl  13078  xrge0tsmsbi  23385  lineval5a  26099  lineval6a  26100  nbcusgra  28170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-sn 3648
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