Elementary ZF axioms inside L

If $a,b\in L$ have the same members in L, transitivity puts all their actual members in L, so ambient Extensionality gives $a=b$. If $a\in L$ is nonempty, ambient Foundation gives $u\in a$ with $u\cap a=\mathrm{\varnothing }$; transitivity puts $u$ in L, so it is also an internal Foundation witness. Empty Set holds because $\mathrm{\varnothing }\in L_1$.

For $a,b\in L$, put both parameters in one nonempty level $L_{\gamma }$. The formula $x=a\vee x=b$ defines $\{a,b\}$ as a subset of this level, hence puts it in $L_{\gamma +1}$. For $a\in L_{\gamma }$, transitivity twice ensures that every member of $\bigcup a$ is in $L_{\gamma }$; the formula $\mathrm{\exists }y\in a(x\in y)$ over that level therefore defines exactly $\bigcup a$. Its successor contains this union, including the empty union. These actual sets satisfy the relativized Pairing and Union axioms.

The actual ordinal $\omega$ belongs to L. It contains empty and, with each $n\in \omega$, the actual successor $n\cup \{n\}$. These finite ordinals belong to L; the pairs and unions used in the successor description are the actual operations by step 2.1. Thus $\omega$ is an internal inductive set, proving Infinity and completing the six asserted axioms.