Power Set is Closed under Set Difference

Theorem
Let $S$ be a set.

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

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

where $A \setminus B$ denotes the set difference of $A$ and $B$.

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 $A \setminus B \subseteq A$ from Set Difference is Subset.

Thus by Subset Relation is Transitive, $A \setminus B \subseteq S$.

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

Also see

 * Power Set is Closed under Intersection
 * Power Set is Closed under Union