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Theorem elsnc 3829
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
Hypothesis
Ref Expression
elsnc.1  |-  A  e. 
_V
Assertion
Ref Expression
elsnc  |-  ( A  e.  { B }  <->  A  =  B )

Proof of Theorem elsnc
StepHypRef Expression
1 elsnc.1 . 2  |-  A  e. 
_V
2 elsncg 3828 . 2  |-  ( A  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:  sneqr  3958  opthwiener  4450  snsn0non  4691  opthprc  4916  dmsnn0  5326  dmsnopg  5332  cnvcnvsn  5338  sniota  5436  funconstss  5839  fniniseg  5842  fniniseg2  5844  fsn  5897  fnse  6454  eusvobj2  6573  fisn  7423  mapfien  7642  axdc3lem4  8322  axdc4lem  8324  axcclem  8326  ttukeylem7  8384  opelreal  8994  seqid3  11355  seqz  11359  1exp  11397  hashf1lem2  11693  imasaddfnlem  13741  0subg  14953  0nsg  14973  sylow2alem2  15240  gsumval3  15502  gsumzaddlem  15514  lsssn0  16012  r0cld  17758  alexsubALTlem2  18067  tgphaus  18134  isusp  18279  i1f1lem  19569  ig1pcl  20086  plyco0  20099  plyeq0lem  20117  plycj  20183  wilthlem2  20840  dchrfi  21027  hsn0elch  22738  h1de2ctlem  23045  atomli  23873  kerunit  24249  kerf1hrm  24250  qqhval2lem  24353  qqhf  24358  qqhre  24374  sibfof  24642  subfacp1lem6  24859  wfrlem14  25524  ellimits  25705  itg2addnclem2  26203  0idl  26572  keridl  26579  smprngopr  26599  isdmn3  26621  pw2f1ocnv  27045  usgra2pthlem1  28184  bnj149  29100  ellkr  29726  diblss  31807  dihmeetlem4preN  31943  dihmeetlem13N  31956
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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