 reserve x,y,X1,X2,X3,X4,X5,X6,Y,Y1,Y2,Y3,Y4,Y5,Z,Z1,Z2,Z3,Z4,Z5 for set;
 reserve X for non empty set;

theorem
  ex Y st Y in X & for Y1 st Y1 in Y holds Y1 misses X
proof
  defpred P[set] means $1 meets X;
  consider Z such that
A1: for Y holds Y in Z iff Y in union X & P[Y] from XFAMILY:sch 1;
  consider Y such that
A2: Y in X \/ Z and
A3: Y misses X \/ Z by Th1;
  assume
A4: not thesis;
  now
    assume
A5: Y in X;
    then consider Y1 such that
A6: Y1 in Y and
A7: not Y1 misses X by A4;
    Y1 in union X by A5,A6,TARSKI:def 4;
    then Y1 in Z by A1,A7;
    then Y1 in X \/ Z by XBOOLE_0:def 3;
    hence contradiction by A3,A6,XBOOLE_0:3;
  end;
  then Y in Z by A2,XBOOLE_0:def 3;
  then Y meets X by A1;
  hence contradiction by A3,XBOOLE_1:70;
end;
