A finite collection of first-order formulas has a closed unbounded class of ordinals whose levels in a definable continuous hierarchy agree with its union for all parameters in the level. Close the collection under subformulas, bound the least witness levels for existential instances on each set-sized stage by Axiom schema of replacement, and iterate these bounds countably. Continuity supplies the closed stages, and induction on formulas proves agreement. The Tarski-Vaught test describes the same witness criterion. For a hierarchy of length an uncountable regular cardinal, bounds stay below that cardinal when every stage has smaller cardinality.
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