Supremum of Power Set

Theorem

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on $\powerset S$ by the relation $\subseteq$.

(From Subset Relation on Power Set is Partial Ordering, this is an ordered set.)

Then the supremum of $\struct {\powerset S, \subseteq}$ is the set $S$.

Proof

By the definition of the power set:

$\forall X \in \powerset S: X \subseteq S$

The result then follows from the definition of supremum.

$\blacksquare$