Club reflection below an inaccessible cardinal

For inaccessible $\kappa$ and any predicate $R\subseteq V_{\kappa }$, the ordinals $\alpha <\kappa$ with $⟨V_{\alpha },\in ,R\cap V_{\alpha }⟩\prec ⟨V_{\kappa },\in ,R⟩$ form a club. Closure under ranks of existential witnesses gives unboundedness, and the elementary chain theorem gives closedness.