Unary intersection of the empty set In MK (Morse-Kelley) set theory life is easy: $\forall X\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)$. If $X=\left\{\right\}$ then $\bigcap X=U$, where $U$ is the universal class. So the (unary) intersection of the empty set is the class that contains all sets as elements. In ZF (Zermelo-Fraenkel) set theory, instead, proper classes are not allowed. So, how can I define $\bigcap X$ in ZF? I tried with the following definitions:


*

*$\forall X\left(X\not=\left\{\right\}\rightarrow\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)\right)$. This means that $\bigcap\left\{\right\}$ is undefined, which is not that good.

*$\forall X\forall y\left(y\in\bigcap X\leftrightarrow X\not=\left\{\right\}\land\forall x\left(x\in X\rightarrow y\in x\right)\right)$. This means that $\bigcap\left\{\right\}=\left\{\right\}$, which is the opposite of MK.


I couldn't find any other valuable definition. Any ideas? Thank you.
 A: 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. 
A: 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. 
A: The only other way I can think of is to use specification, which is really the only way you know that such a set exists.  Given a universe $U$, define $\bigcap X$ to be the unique $B$ satisfying the following formula:
$$\forall x(x\in B\Leftrightarrow(x\in U\wedge \exists y(y\in X\wedge x\in y)))$$
Then $\bigcap\emptyset=U$.  Is this any more or less desirable than having $\bigcap\emptyset$ be a proper class or $\emptyset$ itself?  I don't know.  My topology instructor used it when defining a topology to ensure that the universe was always open, but this is just cosmetic -- there's nothing to be lost by just including this as another axiom.  Ultimately, I suppose everyone but the most hardcore set theorists will just use what works and ignore the axiomatic background.
