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 $$ \begin{equation} a_\alpha |0\rangle = 0 \tag{17} \end{equation} $$