Power Set is Closed under Intersection

Theorem
Let $S$ be a set.

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

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

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

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

From Intersection Subset we have that $A \cap B \subseteq A$.

It follows from Subsets Transitive that $A \cap B \subseteq S$.

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

Also see

 * Power Set Closed under Union
 * Power Set Closed under Set Difference