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