Intersection of Subsemigroups/General Result/Proof 1

Theorem
Let $\struct {S, \circ}$ be a semigroup.

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

Then:

So by Subsemigroup Closure Test, $\struct {T, \circ}$ is a subsemigroup of $\struct {S, \circ}$.

Now to show that $\struct {T, \circ}$ 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 $\forall K \in \mathbb S: K \subseteq T$.

Thus $T$ is the largest subsemigroup of $S$ contained in each member of $\mathbb S$.