Filtered in Meet Semilattice

Theorem
Let $\left({S, \preceq}\right)$ be a meet semilattice.

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

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

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