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

Theorem elsn 3462
Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
Hypothesis
Ref Expression
elsn.1  |-  A  e. 
_V
Assertion
Ref Expression
elsn  |-  ( A  e.  { B }  <->  A  =  B )

Proof of Theorem elsn
StepHypRef Expression
1 elsn.1 . 2  |-  A  e. 
_V
2 elsng 3461 . 2  |-  ( A  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 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sn 3452
This theorem is referenced by:  velsn  3463  sneqr  3604  onsucelsucexmid  4346  ordsoexmid  4378  opthprc  4489  dmsnm  4896  dmsnopg  4902  cnvcnvsn  4907  sniota  5007  fsn  5469  eusvobj2  5638  mapdm0  6420  djulclb  6747  opelreal  7365
  Copyright terms: Public domain W3C validator