On Hereditarily Small Sets in ZF
We show in (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 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 (where the collection under consideration is the set of hereditarily countable sets).
Holmes, M. Randall. (2014). "On Hereditarily Small Sets in ZF". Mathematical Logic Quarterly, 60(3), 228-229. http://dx.doi.org/10.1002/malq.201300089