
theorem Th12:
  for X be set holds {X} is Filter of BoolePoset X
proof
  let X be set;
  now
    let c be object;
    assume c in {X};
    then c = X by TARSKI:def 1;
    then c is Element of BoolePoset X by WAYBEL_8:26;
    hence c in the carrier of BoolePoset X;
  end;
  then reconsider A = {X} as non empty Subset of BoolePoset X by TARSKI:def 3;
  for x,y be set st x c= y & y c= X & x in A holds y in A
  proof
    let x,y be set;
    assume that
A1: x c= y & y c= X and
A2: x in A;
    x = X by A2,TARSKI:def 1;
    then y = X by A1;
    hence thesis by TARSKI:def 1;
  end;
  then
A3: A is upper by WAYBEL_7:7;
  now
    let x,y be set;
    assume x in A & y in A;
    then x = X & y = X by TARSKI:def 1;
    hence x /\ y in A by TARSKI:def 1;
  end;
  hence thesis by A3,WAYBEL_7:9;
end;
