Unary intersection of the empty set

The way I learned it, in ZF, we define the unary union by $$\forall y \left(y\in\cup X \Leftrightarrow \exists z(z\in X\wedge y\in z)\right).$$

The unary intersection is defined using the unary union and the Axiom of Separation: $$\cap X = \left\{ y\in\cup X\,|\, \forall z(z\in X\rightarrow y\in z)\right\}.$$

Using this definition, since $\cup\emptyset = \emptyset$, then $\cap\emptyset=\emptyset$ as well.

The fact that proper classes are handled differently in ZF than in MK doesn't really affect the definition of the unary intersection or other operations that can produce proper classes. So in ZF, $\bigcap z$ is defined using the same definition as in MK: $$ y \in \bigcap z \Leftrightarrow (\forall w \in z) ( y \in w). $$ Just like in MK, $\bigcap \emptyset$ is a proper class in ZF.

There are treatments of ZF in which the authors don't handle proper classes at all, but there are other treatments in which proper classes are "allowed". The simplest way to allow them is to restrict to definable proper classes, and silently replace each such class with its definition whenever the class is used. In this way, for any class or set $z$ we have a (possibly proper) class $\bigcap z$ defined as above. We can prove as a theorem in ZF (just as in MK) is that if $z$ is a nonempty set then there is a set $\hat{z}$ such that $\hat{z} = \bigcap z$.

Addendum: I want to emphasize that this does not conflict with the answer given by Arturo Magidin. The definition I stated here (in which $\bigcap \emptyset$ is a proper class) is the one used by Levy, Basic Set Theory). The definition in the other answer, in which $\bigcap \emptyset = \emptyset$, is used by other authors (I believe). Kunen, Jech, and Halmos all leave $\bigcap \emptyset$ undefined in their well-known texts.