
theorem
  for X1, X2, X3, X4, X5, X6, X7 being set
  holds not (X1 in X2 & X2 in X3 & X3 in X4 & X4 in X5 & X5 in X6 & X6 in X7 &
    X7 in X1)
proof
  let X1, X2, X3, X4, X5, X6, X7 be set;
  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 X6 and
A6: X6 in X7 and
A7: X7 in X1;
  set Z = { X1,X2,X3,X4,X5,X6,X7 };
  for Y being set st Y in Z holds Z meets Y
  proof
    let Y be set such that
A9: Y in Z;
    now
      per cases by A9,ENUMSET1:def 5;
      suppose
A10:     Y = X1;
        take y = X7;
        thus y in Z & y in Y by A7,A10,ENUMSET1:def 5;
      end;
      suppose
A11:    Y = X2;
        take y = X1;
        thus y in Z & y in Y by A1,A11,ENUMSET1:def 5;
      end;
      suppose
A12:    Y = X3;
        take y = X2;
        thus y in Z & y in Y by A2,A12,ENUMSET1:def 5;
      end;
      suppose
A13:    Y = X4;
        take y = X3;
        thus y in Z & y in Y by A3,A13,ENUMSET1:def 5;
      end;
      suppose
A14:    Y = X5;
        take y = X4;
        thus y in Z & y in Y by A4,A14,ENUMSET1:def 5;
      end;
      suppose
A15:    Y = X6;
        take y = X5;
        thus y in Z & y in Y by A5,A15,ENUMSET1:def 5;
      end;
      suppose
A16:    Y = X7;
        take y = X6;
        thus y in Z & y in Y by A6,A16,ENUMSET1:def 5;
      end;
    end;
    hence thesis by XBOOLE_0:3;
  end;
  hence contradiction by XREGULAR:1;
end;
