Cartesian Product Exists and is Unique

Theorem
Let $A$ and $B$ be classes.

Let $A \times B$ be the cartesian product of $A$ and $B$.

Then $A \times B$ exists and is unique.

Proof
Let $A \times B$ be the cartesian product of $A$ and $B$.

Let $\tuple {x, y} \in A \times B$ such that $\tuple {x, y}$ satisfies the Kuratowski ordered pair formulation.

By Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union:
 * $A \times B \subseteq \powerset {\powerset {A \cup B} }$

By Axiom of Specification, there exists a set:
 * $C = \set {\tuple {x, y} \in \powerset {\powerset {A \cup B} }: x \in A \land y \in B}$

By definition, $C$ is the cartesian product of $A$ and $B$.

Thus $A \times B$ exists.

By Cartesian Product is Unique, $A \times B$ is unique.