I'm asked to prove that there is, at best, one equilibrium in a two-player zero sum game where every payoff is different from one another in the matrix representation.
My idea is to suppose there are two equilibria and arrive at some contradiction, but I am having trouble formulating my proof. I was able to prove that these two equilibria would not happen in the same row or column and now I'm trying to develop my proof using the fact that, if an equilibrium happens, one of the players gets $a$ and the other gets $-a$ but, since there is another equilibrium with payoff $b \neq a$, one of them would be more interested in switching strategies (because $b>a$ or $-b > -a$). But this doesn't seem enough to be the contradiction.
Any form of help is appreciated.