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.