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 Th4:
  { X } is Filter of X
proof
A1: for Y1,Y2 holds (Y1 in { X } & Y2 in { X } implies Y1 /\ Y2 in { X }) &
  ( Y1 in { X } & Y1 c= Y2 implies Y2 in { X }) by Th2;
  { X } is non empty Subset-Family of X & not {} in { X } by Th2;
  hence thesis by A1,Def1;
end;
