Supremum of Power Set

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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 an ordered set.)


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.

$\blacksquare$