Axiom Of Choice

We have the following indirect implication of form equivalence classes:

6 \(\Rightarrow\) 6 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
6 \(\Rightarrow\) 6	Zermelo's Axiom of Choice, Moore, [1982]

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

Howard-Rubin Number	Statement
6:	\(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable family of denumerable subsets of \({\Bbb R}\) is denumerable.
6:	\(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable family of denumerable subsets of \({\Bbb R}\) is denumerable.

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