Compact Element iff Principal Ideal

Theorem
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

Let $P = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set

where
 * $\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$,
 * $\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$

Let $x \in \mathit{Ids}\left({L}\right)$

Then $x$ is compact element $x$ is principal ideal in $L$

Proof
By Ideals are Continuous Lattice Subframe of Power Set:
 * $P$ is continuous lattice subframe of $\left({\mathcal P\left({S}\right), \subseteq'}\right)$

where
 * $\mathcal P\left({S}\right)$ denotes the power set of $S$,
 * $\mathord\subseteq' = \mathord\subseteq \cap \left({\mathcal P\left({S}\right) \times \mathcal P\left({S}\right)}\right)$

Sufficient Condition
Assume that
 * $x$ is compact element.

By Compact Element implies Existence of Finite Subset that Element equals Intersection and Includes Subset:
 * $\exists F \in \mathit{Fin}\left({S}\right): x = \bigcap \left\{ {I \in \mathit{Ids}\left({L}\right): F \subseteq I}\right\} \land F \subseteq x$

where $\mathit{Fin}\left({S}\right)$ denotes the set of all finite subsets of $S$.