There Exists No Universal Set/Proof 1

Proof
such a $\UU$ exists.

Using the Axiom of Specification, we can create the set:
 * $R = \set {x \in \UU: x \notin x}$

But from Russell's Paradox, this set cannot exist.

Thus:
 * $R \notin \UU$

and so $\UU$ cannot contain everything.