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We have the following indirect implication of form equivalence classes:

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

Implication	Reference
86-alpha \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 29	Zermelo's Axiom of Choice, Moore, 1982, page 324

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

Howard-Rubin Number	Statement
86-alpha:	\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function.
8:	\(C(\aleph_{0},\infty)\):
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.

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