 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,Y2 st Y1 in Y2 & Y2 in Y holds Y1 misses X
proof
  defpred P[set] means ex Y1 st Y1 in $1 & Y1 meets X;
  consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P[Y] from XFAMILY:sch 1;
  defpred Q[set] means $1 meets X;
  consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & Q[Y] from XFAMILY:sch 1;
  consider Y such that
A3: Y in X \/ Z1 \/ Z2 and
A4: Y misses X \/ Z1 \/ Z2 by Th1;
A5: now
    assume
A6: Y in Z1;
    then consider Y1 such that
A7: Y1 in Y and
A8: Y1 meets X by A1;
    Y in union X by A1,A6;
    then Y1 in union union X by A7,TARSKI:def 4;
    then Y1 in Z2 by A2,A8;
    then Y1 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
    hence contradiction by A4,A7,XBOOLE_0:3;
  end;
  assume
A9: not thesis;
A10: now
    assume
A11: Y in X;
    then consider Y1,Y2 such that
A12: Y1 in Y2 and
A13: Y2 in Y and
A14: not Y1 misses X by A9;
    Y2 in union X by A11,A13,TARSKI:def 4;
    then Y2 in Z1 by A1,A12,A14;
    then Y2 in X \/ Z1 by XBOOLE_0:def 3;
    then Y2 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
    hence contradiction by A4,A13,XBOOLE_0:3;
  end;
  Y in X \/ (Z1 \/ Z2) by A3,XBOOLE_1:4;
  then Y in Z1 \/ Z2 by A10,XBOOLE_0:def 3;
  then Y in Z2 by A5,XBOOLE_0:def 3;
  then Y meets X by A2;
  then Y meets X \/ Z1 by XBOOLE_1:70;
  hence contradiction by A4,XBOOLE_1:70;
end;
