Since we can write the first statement in the equivalent form \begin{equation} \exists \, z \in \mathbb{C}^{N \times 1} \setminus \{ \boldsymbol{0} \} : (A - \lambda I_N) z = \boldsymbol{0} \end{equation} then this result is a subresult to R5081: Complex matrix determinant zero iff some nonzero vector is mapped to zero. $\square$