As you recall from the historical development, Cantor's naive set definition also allowed constructing a "set of all sets". With the axiom of separation, such a set cannot exist.# Corollary: There is no set of all sets
(related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
There is no such set $X$ which contains the set $\{z\in X\mid z\not\in z\}$ as an element. In other words, a "set of all sets" does not exist.
Proofs: 1