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Existence of a set of all sets

Suppose that we have an axiomatic set theory having the following axiom:

**The Axiom Schema of Comprehension:** Let $\mathbf{P}(x)$ be a property of $x$. For any set $A$, there is a set $B$ such that $x\in B$ if and only if $x\in A$ and $\mathbf{P}(x)$.

Can a set of all sets exist within such an axiomatic system?