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Theorem elsn 3513
Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
Hypothesis
Ref Expression
elsn.1  |-  A  e. 
_V
Assertion
Ref Expression
elsn  |-  ( A  e.  { B }  <->  A  =  B )

Proof of Theorem elsn
StepHypRef Expression
1 elsn.1 . 2  |-  A  e. 
_V
2 elsng 3512 . 2  |-  ( A  e.  _V  ->  ( A  e.  { B } 
<->  A  =  B ) )
31, 2ax-mp 5 1  |-  ( A  e.  { B }  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660   {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sn 3503
This theorem is referenced by:  velsn  3514  sneqr  3657  onsucelsucexmid  4415  ordsoexmid  4447  opthprc  4560  dmsnm  4974  dmsnopg  4980  cnvcnvsn  4985  sniota  5085  fsn  5560  eusvobj2  5728  mapdm0  6525  djulclb  6908  opelreal  7603
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