reserve N for Cardinal;
reserve M for Aleph;
reserve X for non empty set;
reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X;
reserve S for Subset-Family of X;
reserve x for set;
reserve F,Uf for Filter of X;
reserve S for non empty Subset-Family of X;
reserve I for Ideal of X;
reserve S,S1 for Subset-Family of X;
reserve FS for non empty Subset of Filters(X);
reserve X for infinite set;
reserve Y,Y1,Y2,Z for Subset of X;
reserve F,Uf for Filter of X;

theorem Th22:
  for F being being_ultrafilter Filter of X for Y holds Y in F iff
  not Y in dual F
proof
  let F be being_ultrafilter Filter of X;
  let Y;
  thus Y in F implies not Y in dual F
  proof
    assume Y in F;
    then not Y` in F by Th6;
    hence thesis by SETFAM_1:def 7;
  end;
  assume not Y in dual F;
  then not Y` in F by SETFAM_1:def 7;
  hence thesis by Def7;
end;
