reserve X for set;
reserve UN for Universe;

theorem Th13:
  X is axiom_GU1 & X is axiom_GU3
  implies
  (for y being set,
       x being Subset of y st y in X holds x in X) &
  (for x,y being set st x c= y & y in X holds x in X)&
  (X is non empty implies {} in X)
  proof
    assume that
A1: for x,y being set st x in X & y in x holds y in X and
A2: for x being set st x in X holds bool x in X;
A3: now
      let y be set;
      let x be Subset of y;
      assume y in X;
      then bool y in X by A2;
      hence x in X by A1;
    end;
    now
      assume X is non empty;
      then consider x be object such that
A4:   x in X;
      reconsider x9 = x as set by TARSKI:1;
      {} c= x9;
      hence {} in X by A4,A3;
    end;
    hence thesis by A3;
  end;
