Grothendieck Universe is Closed under Binary Cartesian Product

Theorem
Let $\mathbb U$ be a Grothendieck universe.

Let $u, v \in \mathbb U$.

Let $u \times v$ be the binary cartesian product of $u$ and $v$ realized as a set of ordered pairs in Kuratowski formalization.

Then $u \times v \in \mathbb U$.

Proof
We have $u \times v \subseteq \map {\PP}{\map{\PP}{u \cup v}}$ by Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union.

Then: