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On hereditarily small sets in ZF

We show in ZF (the usual set theory without Choice) that for any set X, the collection of sets Y such that each element of the transitive closure of {Y} is strictly smaller in size than X (the collection of sets hereditarily smaller than X) is a set. This result has been shown by Jech in the case X=ω1 (where the collection under consideration is the set of hereditarily countable sets).