
theorem Th18:
for X being set st union X = {} holds X = {} or X = {{}}
proof
 let X be set such that
A1: union X = {};
 assume X <> {};
 then consider x being object such that
A2: x in X by XBOOLE_0:def 1;
 thus X = {{}} proof
    thus X c= {{}} proof
      let a be object;
      assume a in X;
      then a = {} by A1,ORDERS_1:6;
      hence thesis by TARSKI:def 1;
    end;
    let a be object;
    assume a in {{}};
    then a = {} by TARSKI:def 1;
    hence a in X by A2,A1,ORDERS_1:6;
  end;
end;
