  In this paper, we provide a positive answer to a question by Matthews whether
$\mathsf{ZF}^-$ is consistent with a non-trivial cofinal Reinhardt elementary
embedding $j\colon V\to V$. The consistency follows from $\mathsf{ZFC} + I_0$,
and more precisely, Schultzenberg's model of $\mathsf{ZF}$ with an elementary
embedding $k\colon V_{\lambda+2}\to V_{\lambda+2}$.
