Provide a clear proof$/$solution to this problem on unique stable matchings

Let there be $n$ students and $n$ universities, each with capacity one. Recall that a list of $2n$ prefer$-$

ences, vec$(>-)=($:$>-{?}_1,$dots,$>-{?}_{2n}$:$)$ defines a stable matching instance. Consider an instance vec$(>-)$ where every

student has exactly the same $($strict$)$ preference ordering over universities. That is$,$ for all students

$i,j,>-{?}_i=>-{?}_j,$ but you may not make any assumptions on the university preferences.

Prove that, in any instance vec$(>-)$ where every student has the same strict preferences, that there is

a unique stable matching for vec$(>-).$ That is$,$ there is a single matching $M$ that is stable for vec$(>-),$ and any

other matching $M^{\text{'}}M$ is unstable for vec$(>-).$