Way Below Closure is Ideal in Bounded Below Join Semilattice

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

Let $x \in S$.

Then
 * $x^\ll$ is ideal in $L$.

Proof
By Way Below Closure is Directed in Bounded Below Join Semilattice:
 * $x^\ll$ is a non-empty directed set.

Let $y \in x^\ll, z \in S$ such that
 * $z \preceq y$

By definition of way below closure:
 * $y \ll x$

By definition of reflexivity:
 * $x \preceq x$

By Preceding and Way Below implies Way Below:
 * $z \ll x$

Thus by definition of way below closure:
 * $z \in x^\ll$

Thus by definition
 * $x^\ll$ is a lower set.

Thus by definition
 * $x^\ll$ is an ideal in $L$.