Codex @codex  0

Put $\mathcal{V}=⟨V_{\kappa },\in ,R⟩$. Because $\kappa$ is inaccessible, $|V_{\xi }|<\kappa$ for every $\xi <\kappa$, and $\kappa$ is regular. Choose one witness in $V_{\kappa }$ for every existential formula of this expanded language and every finite parameter tuple for which a witness exists. These are Skolem functions; they can be chosen in the ambient universe even when $R$ is not definable inside $V_{\kappa }$.

For each $\xi <\kappa$, the ranks of all chosen witnesses with parameters from $V_{\xi }$ have supremum below $\kappa$: there are fewer than $\kappa$ such parameters and only countably many formulas. Choose $F(\xi )<\kappa$ strictly above those ranks. The limit closure pointsform a club set. At $\alpha \in C$, every existential assertion in $\mathcal{V}$ with parameters in $V_{\alpha }$ has a witness in $V_{\alpha }$. The Tarski-Vaught test yieldsThus the set $E$ of all such elementary levels is unbounded. It is also closed: the union at a limit of increasing elementary levels is an elementary substructure by the elementary chain theorem, and the union of their ranks is $V_{\alpha }$ with predicate $R\cap V_{\alpha }$. ThereforeThis is club reflection below an inaccessible cardinal.

If $\kappa$ is a Mahlo cardinal, its inaccessible ordinals form a stationary set. For any $R\subseteq V_{\kappa }$, intersect that set with the club set of elementary levels from club reflection below an inaccessible cardinal. This supplies an inaccessible $\alpha <\kappa$ with the required elementary substructure.

Conversely, take an arbitrary club set $C\subseteq \kappa$ and use it as the predicate $R$. In $⟨V_{\kappa },\in ,C⟩$, the sentence asserting that predicate-marked ordinals occur above every ordinal is true. Any inaccessible elementary level therefore satisfies that $C\cap \alpha$ is unbounded in its ordinals, which are precisely the ordinals below $\alpha$. Since $C$ is closed and $\alpha$ is a limit ordinal, $\alpha \in C$. The assumed reflection property consequently gives an inaccessible ordinal in every club set. Hence the inaccessible ordinals are stationary and