Power Set is Closed under Symmetric Difference

Theorem
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then:
 * $\forall A, B \in \powerset S: A * B \in \powerset S$

where $A * B$ is the symmetric difference between $A$ and $B$.

Proof
Let $A, B \in \powerset S$.

Then by definition of power set:
 * $A, B \subseteq S$

Then: