Power Set is Closed under Union

Theorem
Let $$S$$ be a set.

Let $$\mathcal{P} \left({S}\right)$$ be the power set of $$S$$.

Then:
 * $$\forall A, B \in \mathcal{P} \left({S}\right): A \cup B \in \mathcal{P} \left({S}\right)$$

Proof
Let $$A, B \in \mathcal{P} \left({S}\right)$$.

Then by the definition of power set, $$A \subseteq S$$ and $$B \subseteq S$$.

We also have $$A \cup B \subseteq S \iff A \subseteq S \land B \subseteq S$$ from Union Smallest.

Thus $$A \cup B \in \mathcal{P} \left({S}\right)$$, and closure is proved.