Way Below Closure is Directed in Bounded Below Join Semilattice

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

Let $x \in S$.

Then
 * $x^\ll$ is directed.

Proof
By Bottom is Way Below Any Element:
 * $\bot \ll x$

By definition of way below closure:
 * $\bot \in x^\ll$

Thus by definition:
 * $x^\ll$ is a non-empty set.

Let $y, z \in x^\ll$

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

By Join is Way Below if Operands are Way Below
 * $y \vee z \ll x$

By definition of way below closure:
 * $y \vee z \in x^\ll$

By Join Succeeds Operands:
 * $y \preceq y \vee z$ and $z \preceq y \vee z$

Thus by definition
 * $x^\ll$ is directed.