By hypothesis, we have $\mathbb{P}(E \cap F) = \mathbb{P}(\emptyset) = 0$. Under this assumption, independence requires $0 = \mathbb{P}(E \cap F) = \mathbb{P}(E) \mathbb{P}(F)$ and $\mathbb{P}(F) \mathbb{P}(F) = 0$ is true if and only if $\mathbb{P}(E) = 0$ or $\mathbb{P}(F) = 0$. $\square$