Intersection of Semilattice Ideals is Ideal/Set of Sets

Theorem
Let $\struct {S, \preceq}$ be a bounded below join semilattice.

Let $\II$ be a set of ideals in $\struct {S, \preceq}$.

Then $\bigcap \II$ is an ideal in $\struct {S, \preceq}$.

Non-Empty Set
By Bottom in Ideal:
 * $\forall I \in \II: \bot \in I$

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

By definition of intersection:
 * $\bot \in \bigcap \II$

Hence $\bigcap \II$ is non-empty.

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

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

Let $I \in \II$.

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

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

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

Directed Subset
Let $x, y \in \bigcap \II$.

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

Let $I \in \II$.

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:
 * $x \vee y = \sup \set {x, y}$

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

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:
 * $x \vee y \in \bigcap \II$

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