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Theorem elsncg 3838
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 2444 . 2  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 df-sn 3822 . 2  |-  { B }  =  { x  |  x  =  B }
31, 2elab2g 3086 1  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   {csn 3816
This theorem is referenced by:  elsnc  3839  elsni  3840  snidg  3841  eltpg  3853  eldifsn  3929  elsucg  4650  ltxr  10717  elfzp12  11128  ramcl  13399  nbcusgra  21474  xrge0tsmsbi  24226  elzrhunit  24365  elzdif0  24366  frgrancvvdeqlem1  28421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sn 3822
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