Set System Closed under Intersection is Commutative Semigroup

Theorem
Let $\mathcal{S}$ be a system of sets.

Let $\mathcal{S}$ be such that:


 * $\forall A, B \in \mathcal{S}: A \cap B \in \mathcal{S}$

Then $\left({\mathcal{S}, \cap}\right)$ is a commutative semigroup.

Closure
By definition (above), $\left({\mathcal{S}, \cap}\right)$ is closed.

Associativity
The operation $\cap$ is associative from Intersection is Associative.

Commutativity
The operation $\cap$ is commutative from Intersection is Commutative.

Hence, by definition, the result.