Binary Cartesian Product in Kuratowski Formalization contained in Power Set of Power Set of Union

Theorem
Let $S$ and $T$ be sets.

Let $S \times T$ be the binary cartesian product in Kuratowski Formalization of $S$ and $T$.

Then $S \times T \subseteq \powerset{\powerset{S \cup T}}$.

Proof
Let $x \in S$ and $y \in T$.

We have to show, that $\set{\set{x},\set{x,y}} \in \powerset{\powerset{S \cup T}}$.