Intersection of Semilattice Ideals is Ideal/Set of Sets

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

Let $\mathcal I$ be a set of ideals in $\left({S, \preceq}\right)$.

Then $\bigcap \mathcal I$ is an ideal in $\left({S, \preceq}\right)$.

non-empty set
By Bottom in Ideal:
 * $\forall I \in \mathcal I: \bot \in I$

where $\bot$ denotes the smallest element in $S$.

By definition of intersection:
 * $\bot \in \bigcap \mathcal I$

Hence $\bigcap \mathcal I$ is non-empty.

Lower Set
Let $x \in \bigcap \mathcal I$, $y \in S$ such that
 * $y \preceq x$

We will prove that
 * $\forall I \in \mathcal I: y \in I$

Let $I \in \mathcal I$.

By definition of intersection:
 * $x \in I$

Thus by definition of lower set:
 * $y \in I$

Thus by definition of intersection:
 * $y \in \bigcap \mathcal I$

Directed
Let $x, y \in \bigcap \mathcal I$.

We will prove that
 * $\forall I \in \mathcal I: x \vee y \in I$

Let $I \in \mathcal I$.

By definition of intersection:
 * $x, y \in I$

By definition of directed:
 * $\exists z \in I: x \preceq z \land y \preceq z$

By definition of join:
 * $x \vee y = \sup \left\{ {x, y}\right\}$

By definition:
 * $z$ is upper bound for $\left\{ {x, y}\right\}$.

By definition of supremum:
 * $x \vee y \preceq z$

Thus by definition of lower set:
 * $x \vee y \in I$

Thus by definition of intersection:
 * $x \vee y \in bigcap \mathcal I$

Hence by Directed in Join Semilattice:
 * $\bigcap \mathcal I$ is directed.