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We have the following indirect implication of form equivalence classes:

29 \(\Rightarrow\) 31 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
29 \(\Rightarrow\) 31	Zermelo's Axiom of Choice, Moore, 1982, 31

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

Howard-Rubin Number	Statement
29:	If \(|S| = \aleph_{0}\) and \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets and \(|A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup^{}_{x\in S} A_{x}| = |\bigcup^{}_{x\in S} B_{x}|\). Moore, G. [1982], p 324.
31:	\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem:  The union of a denumerable set of denumerable sets is denumerable.

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