Supremum of Power Set
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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.
$\blacksquare$