Set System Closed under Union is Commutative Semigroup

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

Let $$\mathcal{S}$$ be such that:
 * $$\forall A, B \in \mathcal{S}: A \cup B \in \mathcal{S}$$

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

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

Associativity
The operation $$\cup$$ is associative from Union is Associative.

Commutativity
The operation $$\cup$$ is commutative from Union is Commutative.

Hence, by definition, the result.