Supremum of Power Set

Theorem
Let $$S$$ be a set.

Let $$\mathcal P \left({S}\right)$$ be the power set of $$S$$.

Let $$\left({\mathcal{P} \left({S}\right), \subseteq}\right)$$ be the relational structure defined on $$\mathcal{P} \left({S}\right)$$ by the relation $$\subseteq$$.

(From Subset Relation on Power Set is Partial Ordering, this is a poset.)

Then the supremum of $$\left({\mathcal{P} \left({S}\right), \subseteq}\right)$$ is the set $$S$$.

Proof
By the definition of the power set:
 * $$\forall X \in \mathcal P \left({S}\right): X \subseteq S$$

The result then follows from the definition of supremum.