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.
"The disjunction and related properties for constructive Zermelo-Fraenkel set theory." J. Symbolic Logic 70 (4) 1233 - 1254, December 2005. https://doi.org/10.2178/jsl/1129642124