For the last case, the minus and plus signs apply when the sequence $\alpha ,\alpha_1, \alpha_2, \dots, \alpha_n$ and $\alpha_1', \alpha_2', \dots, \alpha_{n+1}'$ are related to each other via even and odd permutations. If we assume that $\alpha \notin \{\alpha_i\}$ we obtain $$\begin{equation} \langle\alpha_1\alpha_2 \dots \alpha_n|a_\alpha|\alpha_1'\alpha_2' \dots \alpha_{n+1}'\rangle = 0 \tag{15} \end{equation}$$ when $\alpha \in \{\alpha_i'\}$. If $\alpha \notin \{\alpha_i'\}$, we obtain $$\begin{equation} a_\alpha\underbrace{|\alpha_1'\alpha_2' \dots \alpha_{n+1}'}\rangle_{\neq \alpha} = 0 \tag{16} \end{equation}$$ and in particular $$\begin{equation} a_\alpha |0\rangle = 0 \tag{17} \end{equation}$$