There Exists No Universal Set/Proof 3

Proof
Let $\SS$ be the set of all sets.

Then $\SS$ must be an element of itself:
 * $\SS \owns \SS$

Thus we have an infinite descending sequence of membership:


 * $\SS \owns \SS \owns \SS \owns \cdots$

But by No Infinitely Descending Membership Chains, no such sequence exists, a contradiction.