Filtered iff Upper Closure Filtered

Theorem
Let $\left({S, \precsim}\right)$ be a preordered set.

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

Then $H$ is filtered $H^\succsim$ is filtered

where $H^\succsim$ denotes the upper closure of set.

Proof
This follows by mutatis mutandis of the proof of Directed iff Lower Closure Directed.