  $\mathsf{ZF + AD}$ proves that for all nontrivial forcings $\mathbb{P}$ on a
wellorderable set of cardinality less than $\Theta$, $1_{\mathbb{P}}
\Vdash_{\mathbb{P}} \neg\mathsf{AD}$. $\mathsf{ZF + AD} + \Theta$ is regular
proves that for all nontrivial forcing $\mathbb{P}$ which is a surjective image
of $\mathbb{R}$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} \neg\mathsf{AD}$. In
particular, $\mathsf{ZF + AD + V = L(\mathbb{R})}$ proves that for every
nontrivial forcing $\mathbb{P} \in L_\Theta(\mathbb{R})$, $1_{\mathbb{P}}
\Vdash_{\mathbb{P}} \neg\mathsf{AD}$.
