Power Set with Intersection and Superset Relation is Ordered Semigroup

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

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

Then $\struct {\powerset S, \cap, \supseteq}$ 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.

We have that $\supseteq$ is the dual to $\subseteq$.

Hence $\struct {\powerset S, \supseteq}$ is an ordered set.

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

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

Thus:

Hence the result.