Join is Way Below if Operands are Way Below

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

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

where $\ll$ denotes the way below relation.

Then
 * $x \vee y \ll z$

Proof
Let $D$ be a directed subset of $S$ such that
 * $D$ admits a supremum

and
 * $z \preceq \sup D$

By definition of way below relation:
 * $\exists d_1 \in D: x \preceq d_1$

and
 * $\exists d_2 \in D: y \preceq d_2$

By definition of directed subset:
 * $\exists d \in D: d_1 \preceq d$ and $d_2 \preceq d$

By definition of transitivity:
 * $x \preceq d$ and $y \preceq d$

Thus by definition of supremum:
 * $x \vee y \preceq d$

Thus by definition way below relation:
 * $x \vee y \ll z$