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


Proof

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

$\blacksquare$