\(\min_x \max_y f(x,y) \ge \max_y \min_x f(x,y)\).

The proof is almost trivial. Let \((x^*, y^*)\) optimize the left hand side and let \((x_*, y_*)\) optimize the right hand side. Now note two facts: for any given \(x\), \(f(x, \arg\max_y f(x,y)) \ge f(x,y) \forall y\); for any given \(y\), \(f(x, y) \ge f(\arg\min_x f(x,y), y) \forall x\).

So in particular, \(f(x^*, y^*) \ge f(x^*, y_*)\) and \(f(x^*, y_*) \ge f(x_*, y_*)\). So putting everything together, \(f(x^*, y^*) \ge f(x_*,y_*)\).