theorem Th1:
  ex Y st Y in X & Y misses X
proof
  consider x being object such that
A1:  x in X by XBOOLE_0:def 1;
  consider Y such that
A2: Y in X and
A3: not ex x be object st x in X & x in Y by A1,TARSKI:3;
  take Y;
  not ex x being object st x in X & x in Y by A3;
  hence thesis by A2,XBOOLE_0:3;
end;
