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;

theorem Th6:
  Y in F implies not (X \ Y) in F
proof
  assume
A1: Y in F;
  assume X \ Y in F;
  then
A2: Y /\ (X \ Y) in F by A1,Def1;
  Y misses (X \ Y) by XBOOLE_1:79;
  then {} in F by A2;
  hence contradiction by Def1;
end;
