The complex of injective words of permutations which are not derangements is contractible

Let $D_n \subseteq \Sigma_n$ be the set of derangements in the symmetric group. We prove that the complex of injective words generated by $\Sigma_n \setminus D_n$ is contractible. This gives a conceptual explanation to the well known fact that the complex of injective words generated by $\Sigma_n$ is homotopy equivalent to the wedge sum $\underset{|D_n|}{\bigvee} S^{n-1}$.