MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsnc2 Unicode version

Theorem elsnc2 3629
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 3628 . 2  |-  ( B  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 2757   {csn 3600
This theorem is referenced by:  fparlem1  6138  fparlem2  6139  el1o  6452  fin1a2lem11  7990  fin1a2lem12  7991  elnn0  9920  elfzp1  10788  fsumss  12149  elhoma  13812  islpidl  15946  zrhrhmb  16413  rest0  16848  divstgphaus  17753  taylfval  19686  elch0  21779  atoml2i  22909  dibopelvalN  30484  dibopelval2  30486
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 2759  df-sn 3606
  Copyright terms: Public domain W3C validator