I think I've proved the claim, but I'm running into a disagreement with someone.
$[\Rightarrow]:$ Iterated expectations.
$[\Leftarrow]:$ $1_A$ is a nonnegative random variable with 0 mean, hence $1_A=0$ a.s. So $E[1_A|\mathcal F]=0$.
Corollary: $P(A|\mathcal G)=0$ iff $P(A)=0$.
Definitions: $1_A$ is the indicator function on $A$, $\mathcal F$ is some $\sigma$-algebra.