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Theorem elsnc 3839
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 3838 . 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 178    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816
This theorem is referenced by:  sneqr  3968  opthwiener  4460  snsn0non  4702  opthprc  4927  dmsnn0  5337  dmsnopg  5343  cnvcnvsn  5349  sniota  5447  funconstss  5850  fniniseg  5853  fniniseg2  5855  fsn  5908  fnse  6465  eusvobj2  6584  fisn  7434  mapfien  7655  axdc3lem4  8335  axdc4lem  8337  axcclem  8339  ttukeylem7  8397  opelreal  9007  seqid3  11369  seqz  11373  1exp  11411  hashf1lem2  11707  imasaddfnlem  13755  0subg  14967  0nsg  14987  sylow2alem2  15254  gsumval3  15516  gsumzaddlem  15528  lsssn0  16026  r0cld  17772  alexsubALTlem2  18081  tgphaus  18148  isusp  18293  i1f1lem  19583  ig1pcl  20100  plyco0  20113  plyeq0lem  20131  plycj  20197  wilthlem2  20854  dchrfi  21041  hsn0elch  22752  h1de2ctlem  23059  atomli  23887  kerunit  24263  kerf1hrm  24264  qqhval2lem  24367  qqhf  24372  qqhre  24388  sibfof  24656  subfacp1lem6  24873  wfrlem14  25553  ellimits  25757  itg2addnclem2  26259  ftc1anclem3  26284  0idl  26637  keridl  26644  smprngopr  26664  isdmn3  26686  pw2f1ocnv  27110  usgra2pthlem1  28336  bnj149  29308  ellkr  29949  diblss  32030  dihmeetlem4preN  32166  dihmeetlem13N  32179
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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