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Theorem elsnc 3604
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 3603 . 2  |-  ( A  e.  _V  ->  ( A  e.  { B } 
<->  A  =  B ) )
31, 2ax-mp 10 1  |-  ( A  e.  { B }  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   _Vcvv 2740   {csn 3581
This theorem is referenced by:  sneqr  3721  opthwiener  4205  snsn0non  4448  opthprc  4689  dmsnn0  5090  dmsnopg  5096  cnvcnvsn  5102  funconstss  5542  fniniseg  5545  fniniseg2  5547  fsn  5595  fnse  6131  sniota  6217  eusvobj2  6270  fisn  7113  mapfien  7332  axdc3lem4  8012  axdc4lem  8014  axcclem  8016  ttukeylem7  8075  opelreal  8685  seqid3  11021  seqz  11025  1exp  11062  hashf1lem2  11324  imasaddfnlem  13357  0subg  14569  0nsg  14589  sylow2alem2  14856  gsumval3  15118  gsumzaddlem  15130  lsssn0  15632  r0cld  17356  alexsubALTlem2  17669  tgphaus  17726  i1f1lem  18971  ig1pcl  19488  plyco0  19501  plyeq0lem  19519  plycj  19585  wilthlem2  20234  dchrfi  20421  hsn0elch  21752  h1de2ctlem  22059  atomli  22887  subfacp1lem6  23053  wfrlem14  23603  ellimits  23791  0idl  25982  keridl  25989  smprngopr  26009  isdmn3  26031  pw2f1ocnv  26462  bnj149  27919  ellkr  28409  diblss  30490  dihmeetlem4preN  30626  dihmeetlem13N  30639
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 2237
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 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-sn 3587
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