Inaccessible cardinal in a transitive model

Question

If $\mathfrak M$ is a transitive model of ZFC and $\kappa$ is inaccessible, then $\kappa$ is inaccessible in $\mathfrak M$.

Proof. If $\alpha<\kappa$ then since $\mathfrak M \models$ AC, we must have either $(2^{\alpha})^\mathfrak M<\kappa$ or $(2^{\alpha})^\mathfrak M \geq\kappa$ and the letter is impossible since $2^\alpha < \kappa.$ I do not follow here this last statement "since". However I understand that $|\alpha|\leq|\alpha|^\mathfrak M$.

Answer 1

We have an inclusion $(2^\alpha)^\mathfrak{M} \subseteq 2^\alpha$, so if $(2^\alpha)^\mathfrak{M} \geq \kappa$, then $\kappa \leq (2^\alpha)^\mathfrak{M} \leq 2^\alpha$, which is impossible because $\kappa$ is inaccessible.