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 $L$

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