I would like to find a result telling me that two simple closed curves $\alpha$ and $\beta$ (on a non-orientable surface $S$) are in minimal position if and only if there is not a disk in $S$ whose boundary consists of an arc of $\alpha$ and an arc of $\beta$.

[By minimal position I mean that the number of $i(\alpha,\beta) :=$ intersections of $\alpha$ and $\beta$ = min{$i(\alpha',\beta')$|$\alpha'$ and $\beta'$ are isotopic to $\alpha$ and $\beta$ resp.}

I have searched previous literature on non-orientable surfaces and found Lemma 2.5 in: https://www.math.stonybrook.edu/~mlyubich/Archive/Topology/Epstein.pdf

The Lemma states that for two closed curves $\alpha$ and $\beta$ that intersect on an orientable or non-orientable surface, there always exists an ambient isotopy that separates them. This is well known for orientable surfaces, and to obtain the result for non-orientable you can just lift to the orientable double cover and obtain your ambient isotopy there.

Is there a way of extracting the result about minimal position from this lemma? Or is ambient isotopy going to be too weak?