This paper proves that the disjunction property, the numerical existence property. Church's rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom.
As regards the proof technique, it features a self-validating semantics for CZF that combines realizability for extensional set theory and truth.