reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;
reserve F,G for Subset-Family of D;
reserve P for Subset of D;

theorem
  for X being set, F being Subset-Family of X st F = {X} holds
  COMPLEMENT F = {{}}
proof
  let X be set, F be Subset-Family of X such that
A1: F = {X};
  {} c= X;
  then reconsider G = {{}} as Subset-Family of X by ZFMISC_1:31;
  reconsider G as Subset-Family of X;
  for P being Subset of X holds P in G iff P` in F
  proof
    let P be Subset of X;
    hereby
      assume P in G;
      then P = {}X by TARSKI:def 1;
      hence P` in F by A1,TARSKI:def 1;
    end;
    assume P` in F;
    then
A2: P` = [#]X by A1,TARSKI:def 1;
    P = P`` .= {} by A2,XBOOLE_1:37;
    hence thesis by TARSKI:def 1;
  end;
  hence thesis by Def7;
end;
