Existence of Class Intersection

Theorem

For any classes $X, Y$, the intersection $X \cap Y$ exists and is unique.

That is:

$\forall X, Y: \exists! Z: \forall u: u \in Z \iff u \in X \land u \in Y$

where $X \cap Y := Z$.

Proof

Let $X, Y$ be arbitrary.

By Axiom $\text B 2$, there is some class $Z$ such that:

$\forall u: u \in Z \iff u \in X \land u \in Y$

satisfying the existence portion of the theorem.

$\Box$

Now, let some $Z'$ satisfy:

$\forall u: u \in Z' \iff u \in X \land u \in Y$

For each set $u$:

$u \in Z \iff u \in X \land u \in Y \iff u \in Z'$

and thus:

$u \in Z \iff u \in Z'$

Thus, by Axiom of Extension:

$Z = Z'$

satisfying the uniqueness portion of the theorem.

$\blacksquare$