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We have the following indirect implication of form equivalence classes:

420 \(\Rightarrow\) 38 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
420 \(\Rightarrow\) 34	Metric spaces and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002A[2002A], Math. Logic Quart.
34 \(\Rightarrow\) 38	The Axiom of Choice, Jech, [1973b]

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
420:	\(UT(\aleph_0\),\(\aleph_0\),cuf): The union of a denumerable set of denumerable sets is cuf.
34:	\(\aleph_{1}\) is regular.
38:	\({\Bbb R}\) is not the union of a countable family of countable sets.

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