Infimum 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 infimum of $\struct {\powerset S, \subseteq}$ is the empty set $\O$.

Proof

Follows directly from Empty Set is Subset of All Sets and the definition of infimum.

$\blacksquare$