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Ackermann in 1956 proposed an axiomatic set theory.

Reinhard proved that Ackermann's set theory equals ZF

It's clear that Zermelo set theory can be interpreted in Ackermann's set theory minus class comprehension.

Is Ackermann's set theory minus class comprehension also equal to ZF?

In other words: did Reinhardt's proof made an essential use of the class comprehension schema in proving the equivalence between Ackermann's set theory and ZF?

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