Suppose that a set having the properties (i) (ii) and (ii) exists.

Suppose that . Then by property (iii) with
we have . Using property (iii) again with yields
. But this is impossible because property (i) says that and can't both be in . So it must be that .

On the other hand suppose that . Then,as above, by property (iii) we have which leads to a contradiction as before.
Thus it must be that .

However (i) says that one of and must be in . This contradiction shows that the set doesn't exist.