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.

Also see

 * Power Set Closed under Intersection
 * Power Set Closed under Set Difference