Power Set with Union and Subset Relation is Ordered Semigroup

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

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

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

Proof
From Power Set with Union is Commutative Monoid, $\struct {\powerset S, \cup}$ 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 $\cup$.

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

Thus:

Hence the result.