This is Lemma 3.2 of this paper by Juvan, Malnic and Mohar.

Theorem. Let $F$ be a family of pairwise non-homotopic and pairwise disjoint closed curves on a surface $\Sigma$ with $b \geq 0$ boundary components. Then

$|F| \leq \max (1, 3(g_{\Sigma}-1)+2b)$,

where $g_{\Sigma}$ is the genus of $\Sigma$ (orientable or non-orientable).

This is Lemma 3.2 of this paper by Juvan, Malnic and Mohar.

Theorem. Let $F$ be a family of non-null homotopic closed curves on a surface $\Sigma$ with $b \geq 0$ boundary components which are pairwise non-homotopic and pairwise disjoint. Then

$|F| \leq \max (1, 3(g_{\Sigma}-1)+2b)$,

where $g_{\Sigma}$ is the genus of $\Sigma$ (orientable or non-orientable).