This question arose while I was carefully trying to understand Godel's paper proving that the continuum hypothesis is relatively consistent with the usual set theory. He starts from axioms which are pretty much those of Bernays and von Neumann, and among the first few there is an axiom that allows you to form a set from the union of two arbitrary sets: $$ \forall a \forall b \exists c \forall x (x \in c \longleftrightarrow (x \in a) \vee (x = b) ). $$
(There is a good discussion of something very close to these same axioms in the 2009 Bulletin of Symbolic Logic paper by Kanamori titled Bernays and Set Theory. There, this is axiom II(2).)
In words, for any arbitrary sets $a$ and $b$ there exists a set $c$ such that for any $x$, $x \in c$ is equivalent to saying that $x$ is in $a$ or $x$ is the same as $b$ (or both). From this you can prove
$$ \forall a \forall b \exists c \forall x (x \in c \longleftrightarrow (x \in a) \vee (x \in b) ). $$ which lets you form the union of two arbitrary sets.
Having grasped this axiom, and noting that the corresponding axiom for intersection was not present, I then expected to find that one of the later axioms provided a starting point from which you could prove the corresponding statement for intersections, which would be something like
$$ \forall a \forall b \exists c \forall x (x \in c \longleftrightarrow (x \in a) \wedge (x \in b) ). $$
Indeed, there is axiom for construction of classes $$ \forall A \exists B \forall x (x \eta B \longleftrightarrow \lnot x \eta A) $$
(a((3) in the Kanamori paper; $x \eta A$ denotes that set $x$ is in class $A$) that lets you form the complement of a class, and you can get the intersection by taking the complement of the union of the complements of two classes. But this only shows that you can form a class from the intersection of two sets or classes. I could not find anything in the axioms that says that this class is a set.
So...
How, using these axioms, is the notion of a set being formed as the "intersection of two sets" justified?