
theorem Th9:
  for x being set holds {x} is c=directed c=filtered
proof
  let x be set;
  set X = {x};
  hereby
    let Y be finite Subset of X;
    take x;
    union Y c= union X by ZFMISC_1:77;
    hence union Y c= x & x in X by TARSKI:def 1,ZFMISC_1:25;
  end;
  let Y be finite Subset of X;
  take x;
  thus for y be set st y in Y holds x c= y by TARSKI:def 1;
  thus thesis by TARSKI:def 1;
end;
