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;

theorem
  for F being set holds F is Filter of X iff (F is non empty
  Subset-Family of X & not {} in F & for Y1,Y2 holds (Y1 in F & Y2 in F implies
  Y1 /\ Y2 in F) & ( Y1 in F & Y1 c= Y2 implies Y2 in F)) by Def1;
