Intersection of Semilattice Ideals is Ideal

Theorem
Let $\struct {S, \preceq}$ be a meet semilattice.

Let $I_1, I_2$ be ideals in $\struct {S, \preceq}$.

Then $I_1 \cap I_2$ is ideal in $\struct {S, \preceq}$

Directed Subset
Let $x, y \in I_1 \cap I_2$.

By definition of intersection:
 * $x, y \in I_1$ and $x, y \in I_2$

By definition of directed subset:
 * $\exists z_1 \in I_1: x \preceq z_1 \land y \preceq z_1$

and
 * $\exists z_2 \in I_2: x \preceq z_2 \land y \preceq z_2$

By Meet Precedes Operands:
 * $z_1 \wedge z_2 \preceq z_1$ and $z_1 \wedge z_2 \preceq z_2$

By definition of lower set:
 * $z_1 \wedge z_2 \in I_1$ and $z_1 \wedge z_2 \in I_2$

By definition of intersection:
 * $z_1 \wedge z_2 \in I_1 \cap I_2$

By definition of meet:
 * $z_1 \wedge z_2 = \inf \set {z_1, z_2}$

By definition of infimum:
 * $x \preceq z_1 \wedge z_2$ and $y \preceq z_1 \wedge z_2$

Thus by definition:
 * $I_1 \cap I_2$ is directed.

Lower Set
Let $x, y \in S$ such that:
 * $x \in I_1 \cap I_2$ and $y \preceq x$

By definition of intersection:
 * $x \in I_1$ and $x \in I_2$

By definition of lower set:
 * $y \in I_1$ and $y \in I_2$

By definition of intersection:
 * $y \in I_1 \cap I_2$

Thus by definition:
 * $I_1 \cap I_2$ is a lower set.

Non-Empty Set
By assumption:
 * $I_1 \ne \O$ and $I_2 \ne \O$

By definition of non-empty:
 * $\exists x: x \in I_1$ and $\exists y: y \in I_2$

By Meet Precedes Operands:
 * $x \wedge y \preceq x$ and $x \wedge y \preceq y$

By definition of lower set:
 * $x \wedge y \in I_1$ and $x \wedge y \in I_2$

By definition of intersection:
 * $x \wedge y \in I_1 \cap I_2$

Thus by definition:
 * $I_1 \cap I_2$ is non-empty.

Thus by definition:
 * $I_1 \cap I_2$ is ideal in $\struct {S, \preceq}$.