 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
  not ( X1 in X2 & X2 in X3 & X3 in X4 & X4 in X5 & X5 in X1)
proof
  assume that
A1: X1 in X2 and
A2: X2 in X3 and
A3: X3 in X4 and
A4: X4 in X5 and
A5: X5 in X1;
  set Z = { X1,X2,X3,X4,X5 };
A6: for Y st Y in Z holds Z meets Y
  proof
    let Y such that
A7: Y in Z;
    now
      per cases by A7,ENUMSET1:def 3;
      suppose
A8:     Y = X1;
        take y = X5;
        thus y in Z & y in Y by A5,A8,ENUMSET1:def 3;
      end;
      suppose
A9:     Y = X2;
        take y = X1;
        thus y in Z & y in Y by A1,A9,ENUMSET1:def 3;
      end;
      suppose
A10:    Y = X3;
        take y = X2;
        thus y in Z & y in Y by A2,A10,ENUMSET1:def 3;
      end;
      suppose
A11:    Y = X4;
        take y = X3;
        thus y in Z & y in Y by A3,A11,ENUMSET1:def 3;
      end;
      suppose
A12:    Y = X5;
        take y = X4;
        thus y in Z & y in Y by A4,A12,ENUMSET1:def 3;
      end;
    end;
    hence thesis by XBOOLE_0:3;
  end;
  X1 in { X1,X2,X3,X4,X5 } by ENUMSET1:def 3;
  hence contradiction by A6,Th1;
end;
