Implications

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We have the following indirect implication of form equivalence classes:

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

Implication	Reference
113 \(\Rightarrow\) 8	Tychonoff's theorem implies AC, Kelley, J.L. 1950, Fund. Math. Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math.
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
113:	Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.
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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