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.