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;

theorem Th10:
  (for Y holds not (Y in F & Y in dual F)) & for Y holds not (Y in
  I & Y in dual I)
proof
  thus for Y holds not (Y in F & Y in dual F)
  proof
    let Y;
    assume that
A1: Y in F and
A2: Y in dual F;
    Y` in F by A2,SETFAM_1:def 7;
    then
A3: Y` /\ Y in F by A1,Def1;
    Y` misses Y by XBOOLE_1:79;
    then {}X in F by A3;
    hence contradiction by Def1;
  end;
  let Y;
  assume that
A4: Y in I and
A5: Y in dual I;
  Y` in I by A5,SETFAM_1:def 7;
  then Y` \/ Y in I by A4,Def2;
  then [#]X in I by SUBSET_1:10;
  hence contradiction by Def2;
end;
