Infimum 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 infimum of $$\left({\mathcal{P} \left({S}\right), \subseteq}\right)$$ is the empty set $$\varnothing$$.

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