Intersection of Subsemigroups/General Result

Theorem
Let $\left({S, \circ}\right)$ be a semigroup. Let $\mathbb S$ be a set of subsemigroups of $\left({S, \circ}\right)$, where $\mathbb S \ne \varnothing$.

Then the intersection $\bigcap \mathbb S$ of the members of $\mathbb S$ is itself a subsemigroup of $\left({S, \circ}\right)$.

Also, $\bigcap \mathbb S$ is the largest subsemigroup of $\left({S, \circ}\right)$ contained in each member of $\mathbb S$.

Proof
Let $T = \bigcap \mathbb S$.

Let $T_k$ be any element of $\mathbb S$. Then:

So $\left({T, \circ}\right)$ is a subsemigroup of $\left({S, \circ}\right)$.

Now to show that $\left({T, \circ}\right)$ is the largest such subsemigroup.

Let $x, y \in T$.

Then $\forall K \subseteq T: x \circ y \in K \implies x \circ y \in T$.

Thus any $K \in \mathbb S: K \subseteq T$ and thus $T$ is the largest subsemigroup of $S$ contained in each member of $\mathbb S$.