Axiom Of Choice

We have the following indirect implication of form equivalence classes:

86-alpha \(\Rightarrow\) 418 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
86-alpha \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 418	Metric spaces and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002A[2002A], Math. Logic Quart.

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:	\(C(\aleph_{0},\infty)\):
418:	DUM(\(\aleph_0\)): The countable disjoint union of metrizable spaces is metrizable.

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