I am trying to prove the existence of mixed-strategy NE for 2-player zero-sum symmetric game, under the condition that given they have $I$ pure strategies and for the pay-off matrix $A$, $\exists x\in R_+^I$ such that $xA\ge0$. I know that I can use Nash's theorem directly, but I want to prove it explicitly.

So far what I have done is that I proved the payoff must be 0 since $A=-A^T$. And if $xA$ always has negative element(s), the best response of each players must be some pure strategy (by decomposing $xAy$). But it seems that all these are irrelevant to a direct proof the existence of NE. Please share with me if you have any idea!