Showing that $\alpha$ isn't a cardinal in $J_{\alpha+1}^{\vec E}$ for a fine extender sequence $\vec E$
We know that $\alpha = (\nu^{+})^{Ult(J^{\vec{E}}_{\alpha}, E_{\alpha})}$ and that $i_{E_{\alpha}} (\kappa) > \nu$, where $i_{E_{\alpha}}$ denotes the ultrapower embedding. Thus working in $Ult(J^{E_{\alpha}}_{\alpha}, E_{\alpha})$ any $\beta <\nu^{+}=\alpha$ can be represented in the ultrapower $Ult(J^{\vec{E}}_{\alpha}, E_{\alpha})$ using a function $f: [\kappa]^{|a|} \rightarrow \kappa$, for some $a \in [\nu]^{<\omega}$ and $f \in J^{E_{\alpha}}_{\alpha}$. Thus we have a surjection from $(P(\kappa) \cap J^{E_{\alpha}}_{\alpha}) \times [\nu]^{<\omega}$ onto $\alpha$, via just looking at the representatives of ordinals less than $\alpha$ in the ultrapower.But the ultrapower can be constructed in $J^{\vec{E}}_{\alpha +1}$ as it has all the information. So $\alpha$ can not be a cardinal in $J^{\vec{E}}_{\alpha +1}$.