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Theorem elsnc2 3835
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that  B, rather than  A, be a set. (Contributed by NM, 12-Jun-1994.)
Hypothesis
Ref Expression
elsnc2.1  |-  B  e. 
_V
Assertion
Ref Expression
elsnc2  |-  ( A  e.  { B }  <->  A  =  B )

Proof of Theorem elsnc2
StepHypRef Expression
1 elsnc2.1 . 2  |-  B  e. 
_V
2 elsnc2g 3834 . 2  |-  ( B  e.  _V  ->  ( A  e.  { B } 
<->  A  =  B ) )
31, 2ax-mp 8 1  |-  ( A  e.  { B }  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806
This theorem is referenced by:  fparlem1  6437  fparlem2  6438  el1o  6734  fin1a2lem11  8279  fin1a2lem12  8280  elnn0  10212  elfzp1  11086  fsumss  12507  elhoma  14175  islpidl  16305  zrhrhmb  16780  rest0  17221  divstgphaus  18140  taylfval  20263  elch0  22744  atoml2i  23874  fprodss  25263  climrec  27643  dibopelvalN  31780  dibopelval2  31782
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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