Using the Pauli principle $$ \begin{equation} |\alpha_1\dots \alpha_i\dots \alpha_i\dots \alpha_n\rangle_{\mathrm{AS}} = 0 \tag{6} \end{equation} $$ it follows that $$ \begin{equation} a_{\alpha_i}^{\dagger} a_{\alpha_i}^{\dagger} = 0. \tag{7} \end{equation} $$ If we combine Eqs. (5) and (7), we obtain the well-known anti-commutation rule $$ \begin{equation} a_{\alpha}^{\dagger} a_{\beta}^{\dagger} + a_{\beta}^{\dagger} a_{\alpha}^{\dagger} \equiv \{a_{\alpha}^{\dagger},a_{\beta}^{\dagger}\} = 0 \tag{8} \end{equation} $$