We define the relative constructible sets, $L[A](\delta)$ as follows:
- $L[A](0)=\{A\}\cup \operatorname{trcl}(A)$
- $L[A](\beta+1)=\mathcal{D}^+(L[A](\beta))$
- $L[A](\gamma)=\bigcup_{\alpha<\gamma} L[A](\alpha)$ for limit ordinals $\gamma$.
and $L[A] = \bigcup_{\alpha \in \mathbf{ON}}L[A](\alpha)$. Note that as mentioned by Asaf here, the conventional notation for this definition should be $L(A)$ instead (for the context of this question, I will stick to $L[A]$).
Exercise II.6.30 afterwards then states that:
For any $A$, $L[A]$ is a transitive model of $\mathsf{ZF}$, and $A \in L[A]$. Also, $L[A] \models \mathsf{AC}$ if $A$ is a set of ordinals, and $L[A] \models \mathsf{GCH}$ if $A \subseteq \omega$.
Both transitivity and $A \in L[A]$ are clear, but I'm not certain about the last two parts. For $\mathsf{AC}$, I have not worked out the details, but I think this follows from that $L[A](0)$ is a well-orderable set. Then the proof of which $L[A] \models \mathsf{AC}$ should follow similarly to that of the original $L \models \mathsf{AC}$. Unfortunately for $L[A] \models \mathsf{GCH}$, I don't have much idea.
Any help is appreciated.