A simple proof for a case where: $\mathbf{L}_\mu \models ZF^-$?

I am looking for a simple proof (no fine structure, please) of the following:

Let $ \lambda$ be a limit ordinal, and $ \mu < \lambda$ , infinite: If $ \mathbf{L}_\lambda \models \texttt{“}\mu \mbox{ is a successor cardinal}\texttt{“}$ then $ \mathbf{L}_\mu \models ZF^-$ .

Where $ \mathbf{L}_\lambda \models \texttt{“}\alpha \mbox{ is a cardinal}\texttt{“}$ , means here: there is no surjection $ \xi \to \alpha$ , in $ \mathbf{L}_\lambda$ , for $ \xi < \alpha$ .

And $ ZF^-$ is Zermelo-Fraenkel minus Power Set.

The proof I have uses some standard long-winded “condensation” arguments + Admissible sets. So I welcome any ideas.

Short of this, can anyone suggest a short, simple proof of the following:

Let $ \lambda$ be a limit ordinal [or even limit of limits]: for all $ x \in \mathbf{L}_\lambda$ , there is in $ \mathbf{L}_\lambda$ a surjection $ \xi \to x$ , where $ \xi <\lambda$ .

Note: Devlin shows in Ch. B.5 of the Handbook of Mathematical Logic: For every limit $ \alpha$ , there is a $ \mathbf{\Sigma}_1(\mathbf{L}_\alpha)$ surjection $ \alpha \to \mathbf{L}_\alpha$

I am trying to avoid using this. Arguments using admissible sets are most definitely ok.

A reference to a published proof would be excellent!