Power Set with Intersection and Subset Relation is Ordered Semigroup

Theorem
Let $S$ be a set and let $\powerset S$ be its power set.

Let $\struct {\powerset S, \cap, \subseteq}$ be the ordered structure formed from the set intersection operation and subset relation.

Then $\struct {\powerset S, \cap, \subseteq}$ is an ordered semigroup.

Proof
From Power Set with Intersection is Commutative Monoid, $\struct {\powerset S, \cap}$ is a semigroup.

From Subset Relation is Ordering, $\struct {\powerset S, \subseteq}$ is an ordered set.

It remains to be shown that $\subseteq$ is compatible with $\cap$.

Let $A, B \subseteq S$ be arbitrary such that $A \subseteq B$.

Thus:

Hence the result.