Filtered in Meet Semilattice

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

Let $H$ be a non-empty upper section of $S$.

Then $H$ is filtered
 * $\forall x, y \in H: x \wedge y \in H$

Proof
This follows by of the proof of Directed in Join Semilattice.