Join Semilattice Ideal iff Ordered Set Ideal

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

Let $I \subseteq S$ be a non-empty subset of $S$.

Then:
 * $I$ is a (join semilattice) ideal of $\struct {S, \vee, \preceq}$ $I$ is an (ordered set) ideal of $\struct {S, \preceq}$.