In the notes I'm following to understand the proof that $L \vDash GCH$, there is this crucial following passage:
"Let $A \subseteq X$. There is some $\alpha$ such that $A ∈ L_\alpha$ , and it suffices to show that the least such $\alpha$ is strictly smaller than $\kappa^+$ . Because then $\mathcal{P}(X) \subseteq L_{\kappa^+}$"
It is probably very simple, but why every $A \in L_\alpha$ where $\alpha < \kappa^+$ implies that $P(\kappa) \subseteq L_{\kappa^+}$? My doubt is because $sup\{\alpha: \exists A \subseteq \kappa \; A \in L_\alpha\}$ is not necessarily less than $\kappa^+$ if the $\alpha$ are.
Does $L$ "see" $|L_\alpha| = |\alpha|$?