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Theorem elsn2 3405
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
elsn2.1  |-  B  e. 
_V
Assertion
Ref Expression
elsn2  |-  ( A  e.  { B }  <->  A  =  B )

Proof of Theorem elsn2
StepHypRef Expression
1 elsn2.1 . 2  |-  B  e. 
_V
2 elsn2g 3404 . 2  |-  ( B  e.  _V  ->  ( A  e.  { B } 
<->  A  =  B ) )
31, 2ax-mp 7 1  |-  ( A  e.  { B }  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 98    = wceq 1243    e. wcel 1393   _Vcvv 2557   {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381
This theorem is referenced by:  el1o  6020  elnn0  8183
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