User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-35470197-20190913130333

By the way, what is the meta theory? If you intend ZFC and x is regarded as an ordinal in the meta theory (but not the formalised KP set theories), then the condition "for all (Goedel number of) statement \(\phi\) such that \(L_x \models \phi\)" works. What I wrote above is under the assumption that you do not intend a specific meta theory.