Does any sequence with values in a directed set have a monotone subsequence?

Definition: a directed set is a set $$M$$ together with a preorder $$\geq$$ (reflexive and transitive order) such that every pair of elements in $$M$$ has an upper bound ($$\forall x,y \in M, \ \exists z \in M,\ z \geq x \wedge z \geq y$$)

For sequences in the reals, there always exists a monotone subsequence. The proof of this fact uses the total order property of the real numbers. But I was wondering, would it work for directed sets? In other words, does a sequence with values in a directed set $$(M, \geq)$$ always have a monotone subsequence?

I tried to find a counterexample, but I have yet to work with directed sets to be able to visualize them at this level.

Let $$M=\wp(\Bbb N)$$, directed by inclusion, and let $$a_n=\{n\}$$ for $$n\in\Bbb N$$; then $$\langle a_n:n\in\Bbb N\rangle$$ is a sequence in $$M$$ with no monotone subsequence.