Ideals are Continuous Lattice Subframe of Power Set

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

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

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

Then $I$ is continuous lattice subframe of $P$.