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Theorem elsnc2 3682
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 3681 . 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 176    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653
This theorem is referenced by:  fparlem1  6234  fparlem2  6235  el1o  6514  fin1a2lem11  8052  fin1a2lem12  8053  elnn0  9983  elfzp1  10852  fsumss  12214  elhoma  13880  islpidl  16014  zrhrhmb  16481  rest0  16916  divstgphaus  17821  taylfval  19754  elch0  21849  atoml2i  22979  climrec  27832  dibopelvalN  31955  dibopelval2  31957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sn 3659
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