 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,Y4,Y5 st Y1 in Y2 &  Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y
    holds Y1 misses X
proof
  defpred P[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
  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 T[set] means $1 meets X;
  defpred S[set] means ex Y1 st Y1 in $1 & Y1 meets X;
  defpred R[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
  defpred Q[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 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 Z5 such that
A3: for Y holds Y in Z5 iff Y in union union union union union X & T[Y]
  from XFAMILY:sch 1;
  consider Z3 such that
A4: for Y holds Y in Z3 iff Y in union union union X & R[Y] from
  XFAMILY:sch 1;
  consider Z4 such that
A5: for Y holds Y in Z4 iff Y in union union union union X & S[Y] from
  XFAMILY:sch 1;
  set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5;
  consider Y such that
A6: Y in V and
A7: Y misses V by Th1;
A8: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:4
    .= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 by XBOOLE_1:4
    .= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 by XBOOLE_1:4
    .= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) by XBOOLE_1:4;
A9: now
    assume
A10: Y in Z1;
    then consider Y1,Y2,Y3,Y4 such that
A11: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A12: Y4 in Y and
A13: Y1 meets X by A1;
    Y in union X by A1,A10;
    then Y4 in union union X by A12,TARSKI:def 4;
    then Y4 in Z2 by A2,A11,A13;
    then Y4 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
    then Y meets X \/ Z1 \/ Z2 by A12,XBOOLE_0:3;
    then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
    then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
    hence contradiction by A7,XBOOLE_1:70;
  end;
A14: now
    assume
A15: Y in Z2;
    then consider Y1,Y2,Y3 such that
A16: Y1 in Y2 & Y2 in Y3 and
A17: Y3 in Y and
A18: Y1 meets X by A2;
    Y in union union X by A2,A15;
    then Y3 in union union union X by A17,TARSKI:def 4;
    then Y3 in Z3 by A4,A16,A18;
    then Y3 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
    then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
    then Y3 in V by XBOOLE_0:def 3;
    hence contradiction by A7,A17,XBOOLE_0:3;
  end;
A19: now
    assume
A20: Y in Z3;
    then consider Y1,Y2 such that
A21: Y1 in Y2 and
A22: Y2 in Y and
A23: Y1 meets X by A4;
    Y in union union union X by A4,A20;
    then Y2 in union union union union X by A22,TARSKI:def 4;
    then Y2 in Z4 by A5,A21,A23;
    then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
    then Y2 in V by XBOOLE_0:def 3;
    hence contradiction by A7,A22,XBOOLE_0:3;
  end;
A24: now
    assume
A25: Y in Z4;
    then consider Y1 such that
A26: Y1 in Y and
A27: Y1 meets X by A5;
    Y in union union union union X by A5,A25;
    then Y1 in union union union union union X by A26,TARSKI:def 4;
    then Y1 in Z5 by A3,A27;
    then Y1 in V by XBOOLE_0:def 3;
    hence contradiction by A7,A26,XBOOLE_0:3;
  end;
  assume
A28: not thesis;
  now
    assume
A29: Y in X;
    then consider Y1,Y2,Y3,Y4,Y5 such that
A30: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A31: Y5 in Y and
A32: not Y1 misses X by A28;
    Y5 in union X by A29,A31,TARSKI:def 4;
    then Y5 in Z1 by A1,A30,A32;
    then Y5 in X \/ Z1 by XBOOLE_0:def 3;
    then Y5 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
    then Y5 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
    then Y meets X \/ Z1 \/ Z2 \/ Z3 by A31,XBOOLE_0:3;
    then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
    hence contradiction by A7,XBOOLE_1:70;
  end;
  then Y in Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A8,A6,XBOOLE_0:def 3;
  then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 \/ Z5 by XBOOLE_1:4;
  then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) \/ Z5 by XBOOLE_1:4;
  then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5) by XBOOLE_1:4;
  then Y in Z2 \/ Z3 \/ Z4 \/ Z5 by A9,XBOOLE_0:def 3;
  then Y in Z2 \/ (Z3 \/ Z4) \/ Z5 by XBOOLE_1:4;
  then Y in Z2 \/ (Z3 \/ Z4 \/ Z5) by XBOOLE_1:4;
  then Y in Z3 \/ Z4 \/ Z5 by A14,XBOOLE_0:def 3;
  then Y in Z3 \/ (Z4 \/ Z5) by XBOOLE_1:4;
  then Y in Z4 \/ Z5 by A19,XBOOLE_0:def 3;
  then Y in Z5 by A24,XBOOLE_0:def 3;
  then Y meets X by A3;
  hence contradiction by A8,A7,XBOOLE_1:70;
end;
