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;

theorem Th15:
  for S being set holds S in Filters(X) iff S is Filter of X
proof
  let S be set;
  thus S in Filters(X) implies S is Filter of X
  proof
    defpred P[set] means $1 is Filter of X;
    assume S in Filters(X);
    then
A1: S in {S1: P[S1]};
    thus P[S] from ElemProp(A1);
  end;
  assume S is Filter of X;
  hence thesis;
end;
