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

Theorem snmg 3535
Description: The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
snmg  |-  ( A  e.  V  ->  E. x  x  e.  { A } )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem snmg
StepHypRef Expression
1 snidg 3450 . 2  |-  ( A  e.  V  ->  A  e.  { A } )
2 elex2 2628 . 2  |-  ( A  e.  { A }  ->  E. x  x  e. 
{ A } )
31, 2syl 14 1  |-  ( A  e.  V  ->  E. x  x  e.  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1424    e. wcel 1436   {csn 3425
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-sn 3431
This theorem is referenced by:  snm  3537  prmg  3538  xpimasn  4836  1stconst  5924  2ndconst  5925
  Copyright terms: Public domain W3C validator