 In the case of CZF, the claim that adding the power set axiom  induces a form of impredicativity can be substantiated by a technical result. Rathjen (2012b) shows that CZF augmented by the power set axiom exceeds the strength of classical Zermelo set theory, and thus the addition of the power set axiom to CZF  brings us to a fully impredicative theory. This also shows that the implication from power set to subset collection can not be reversed, as CZF’s proof-theoretic strength is way below that of Zermleo set-theory. In other terms, the power set axiom is much stronger than  both exponentiation and subset collection. 