# 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!