Prime Ideal is Prime Element

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice.

Let $I$ be an ideal in $L$.

Then
 * $I$ is a prime ideal


 * $I$ is a prime element in $\left({ \mathit{Ids}\left({L}\right), \precsim}\right)$

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

Sufficient Condition
Let $I$ be a prime ideal.

Let $x, y \in \mathit{Ids}\left({L}\right)$ such that
 * $x \wedge y \precsim I$

By definition of $\precsim$:
 * $x \wedge y \subseteq I$

By Intersection of Ideals is Ideal and Meet in Set of Ideals:
 * $x \cap y \subseteq I$

Aiming for a contraindication suppose
 * $x \not\precsim I$ and $y \not\precsim I$