Inaccessible cardinal and well founded set

Question

How can I prove the following fact? Let be a regular cardinal $\kappa$ such that $V_\kappa$ is a model of ZFC. Then $\kappa$ is inaccessible.

Answer 1

Since we assume $\kappa$ to be regular, we only need to prove that for any $\lambda < \kappa$ we have $2^\lambda < \kappa$. Note that for every ordinal $\alpha$ we have $\operatorname{rank}(\alpha) = \alpha$ (see, e.g. this answer). In particular, that means that the collection of ordinals in $V_\kappa$ is just $\kappa$. So for $\lambda < \kappa$, we have $\lambda \in V_\kappa$ and all of its subsets belong to $V_\kappa$. Then since $V_\kappa$ is a model of ZFC, the powerset $\mathcal{P}(\lambda)$ must belong to $V_\kappa$ again. Thus $2^\lambda$ (as an ordinal) is also in $V_\kappa$, and so we see $2^\lambda < \kappa$.