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 Th7:
  for I being non empty Subset-Family of X st (for Y holds Y in I
iff Y` in F) holds (not X in I) & for Y1,Y2 holds (Y1 in I & Y2 in I implies Y1
  \/ Y2 in I) & ( Y1 in I & Y2 c= Y1 implies Y2 in I)
proof
  let I be non empty Subset-Family of X such that
A1: for Y holds (Y in I iff Y` in F);
  not ({}X)`` in F by Def1;
  hence not X in I by A1;
  let Y1,Y2;
  thus Y1 in I & Y2 in I implies Y1 \/ Y2 in I
  proof
    assume Y1 in I & Y2 in I;
    then Y1` in F & Y2` in F by A1;
    then Y1` /\ Y2` in F by Def1;
    then (Y1 \/ Y2)` in F by XBOOLE_1:53;
    hence thesis by A1;
  end;
  assume that
A2: Y1 in I and
A3: Y2 c= Y1;
A4: Y1` c= Y2` by A3,SUBSET_1:12;
  Y1` in F by A1,A2;
  then Y2` in F by A4,Def1;
  hence thesis by A1;
end;
