Power Set is Boolean Ring

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

Denote with $*$ and $\cap$ symmetric difference and intersection, respectively.

Then $\left({S, *, \cap}\right)$ is a Boolean ring.

Proof
From Symmetric Difference with Intersection forms Ring, $\left({S, *, \cap}\right)$ is a ring with unity.

By Intersection is Idempotent, $\cap$ is idempotent.

It follows that $\left({S, *, \cap}\right)$ is a Boolean ring.