Prove that a subclass of any set is also a set

Question

I have been asked, while discussing the ZF subset axioms to prove that a subclass of any set is a set. I have put together the following proof and would be grateful if someone could confirm that it is correct.

By definition of classes, we know that classes and subclasses are given by predicates. Using the language of our axiom, whenever we have a set $A$ along with a predicate $\phi$, there exists a set $B$ which contains exactly the elements of $A$ which satisfy $\phi$. Hence if $A$ is a set then so must be the class $B=\{a\in A \ | \ \phi (a)\}$ which contains all elements of A which satisfy $\phi$.

Answer 1

This is exactly true because a class is a predicate and the axiom of separation holds by assumption: it tells us that the intersection of a set and a class is a set, in these terms.