Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E, F \in \mathcal{F}$ are each an D1716: Event in $P$
(ii)	\begin{equation} E \cap F = \emptyset \end{equation}

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 $\mathbb{P}(F) = 0$. $\square$