 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,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in Y holds Y1 misses X
proof
  defpred P[set] means ex Y1,Y2 st Y1 in Y2 & Y2 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 R[set] means $1 meets X;
  defpred Q[set] means ex Y1 st Y1 in $1 & Y1 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 Z3 such that
A3: for Y holds Y in Z3 iff Y in union union union X & R[Y]
    from XFAMILY:sch 1;
  consider Y such that
A4: Y in X \/ Z1 \/ Z2 \/ Z3 and
A5: Y misses X \/ Z1 \/ Z2 \/ Z3 by Th1;
A6: now
    assume
A7: Y in Z2;
    then consider Y1 such that
A8: Y1 in Y and
A9: Y1 meets X by A2;
    Y in union union X by A2,A7;
    then Y1 in union union union X by A8,TARSKI:def 4;
    then Y1 in Z3 by A3,A9;
    then Y1 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
    hence contradiction by A5,A8,XBOOLE_0:3;
  end;
A10: now
    assume
A11: Y in Z1;
    then consider Y1,Y2 such that
A12: Y1 in Y2 and
A13: Y2 in Y and
A14: Y1 meets X by A1;
    Y in union X by A1,A11;
    then Y2 in union union X by A13,TARSKI:def 4;
    then Y2 in Z2 by A2,A12,A14;
    then Y2 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
    then Y meets X \/ Z1 \/ Z2 by A13,XBOOLE_0:3;
    hence contradiction by A5,XBOOLE_1:70;
  end;
  set V = X \/ Z1 \/ Z2 \/ Z3;
A15: V = X \/ (Z1 \/ Z2) \/ Z3 by XBOOLE_1:4
    .= X \/ (Z1 \/ Z2 \/ Z3) by XBOOLE_1:4;
  assume
A16: not thesis;
  now
    assume
A17: Y in X;
    then consider Y1,Y2,Y3 such that
A18: Y1 in Y2 & Y2 in Y3 and
A19: Y3 in Y and
A20: not Y1 misses X by A16;
    Y3 in union X by A17,A19,TARSKI:def 4;
    then Y3 in Z1 by A1,A18,A20;
    then Y3 in X \/ Z1 by XBOOLE_0:def 3;
    then Y3 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
    then Y3 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
    hence contradiction by A5,A19,XBOOLE_0:3;
  end;
  then Y in Z1 \/ Z2 \/ Z3 by A15,A4,XBOOLE_0:def 3;
  then Y in Z1 \/ (Z2 \/ Z3) by XBOOLE_1:4;
  then Y in Z2 \/ Z3 by A10,XBOOLE_0:def 3;
  then Y in Z3 by A6,XBOOLE_0:def 3;
  then Y meets X by A3;
  hence contradiction by A15,A5,XBOOLE_1:70;
end;
