Suppose that is a regular cardinal such that for all . Prove that . Which axioms of ZFC does satisfy?

Here, for a given cardinal , denotes the collection of sets whose transitive closure has cardinality less than . Also, a well orderable infinite cardinal is regular if , where denotes the cofinality. The set is defined by , , and if .

I am not sure how to prove this. I would appreciate a few hints or suggestions. Thank you.

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Zermelo?Fraenkel set theory - Wikipedia, the free encyclopedia