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
By Law of Excluded Middle there are two choices:
 * $S = \O \text{ or } T = \O$
 * $\neg \paren {S = \O \text{ or } T = \O}$

Suppose $S = \O \text{ or } T = \O$.

Suppose $S \times T$ is non-empty.

Then there exists some ordered pair $\tuple {x, y}$ where:
 * $\tuple {x, y} \in S \times T$

By the definition of cartesian products:
 * $x \in S$
 * $y \in T$

Then $S$ and $T$ are both non-empty.

But this contradicts our assumption that one of them is empty.

Suppose $\neg \paren{S = \O \text{ or } T = \O}$.

By De Morgan's Laws:
 * $S \ne \O \text{ and } T \ne \O$

Thus there exist $x$ and $y$ such that:
 * $x \in S$
 * $y \in T$

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