Compact Element iff Existence of Finite Subset that Element equals Intersection and Includes Subset

Theorem
Let $X, E$ be sets.

Let $P = \left({\mathcal P\left({S}\right), \precsim}\right)$ be an inclusion ordered set

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

Let $L = \left({S, \preceq}\right)$ be a continuous lattice subframe of $P$.

Then $E$