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Theorem elsncg 3636
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
StepHypRef Expression
1 eqeq1 2264 . 2  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 df-sn 3620 . 2  |-  { B }  =  { x  |  x  =  B }
31, 2elab2g 2891 1  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   {csn 3614
This theorem is referenced by:  elsnc  3637  elsni  3638  snidg  3639  eltpg  3650  eldifsn  3723  elsucg  4431  ltxr  10425  elfzp12  10828  ramcl  13039  lineval5a  25456  lineval6a  25457
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-sn 3620
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