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 Cartesian product of $S$ and $T$ realized as a set of ordered pairs in Kuratowski formalization.

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

Proof
By Law of Excluded Middle there are two choices:
 * $S \times T = \O$
 * $S \times T \ne \O$

Suppose $S \times T = \O$.

By Empty Set is Subset of Power Set:
 * $S \times T \subseteq \powerset {\powerset {S \cup T} }$

Suppose $S \times T \ne \O$.

By Cartesian Product is Empty iff Factor is Empty, there exist $x$ and $y$ such that:
 * $x \in S$
 * $y \in T$

Let us express the ordered pair $\tuple {x, y}$ using the Kuratowski formalization:

$\tuple {x, y} \equiv \set {x, \set {x, y} }$

We now show that:
 * $\set {\set x, \set {x, y} } \in \powerset {\powerset {S \cup T} }$

Indeed: