If Element Does Not Belong to Ideal then There Exists Prime Ideal Including Ideal and Excluding Element

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

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

Let $x$ be an element of $S$.

Suppose $x \notin I$

Then there exists a prime ideal $P$ in $L$: $I \subseteq P$ and $x \notin P$