Identity of Power Set with Intersection

Theorem
Let $S$ be a set and let $\mathcal P \left({S}\right)$ be its power set.

Consider the algebraic structure $\left({\mathcal P \left({S}\right), \cap}\right)$, where $\cap$ denotes set intersection.

Then $S$ serves as the identity for $\left({\mathcal P \left({S}\right), \cap}\right)$.

Proof
We note that from Subset of Itself, $S \subseteq S$ and so $S \in \mathcal P \left({S}\right)$ from the definition of the power set.

From Intersection with Subset is Subset‎, we have:
 * $A \subseteq S \iff A \cap S = A = S \cap A$

By definition of power set:
 * $A \subseteq S \iff A \in \mathcal P \left({S}\right)$

So:
 * $\forall A \in \mathcal P \left({S}\right): A \cap S = A = S \cap A$

Thus we see that $S$ acts as the identity element.