ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elsn2 Unicode version

Theorem elsn2 3564
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 3563 . 2  |-  ( B  e.  _V  ->  ( A  e.  { B } 
<->  A  =  B ) )
31, 2ax-mp 5 1  |-  ( A  e.  { B }  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1332    e. wcel 1481   _Vcvv 2689   {csn 3530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3536
This theorem is referenced by:  el1o  6340  elnn0  9001  elxnn0  9064  fisumss  11191  rest0  12380
  Copyright terms: Public domain W3C validator