Implications

We have the following indirect implication of form equivalence classes:

113 \(\Rightarrow\) 421
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 \(\Rightarrow\) 421 Unions and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002B[2002B], Math. Logic Quart.

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:

\(C(\aleph_{0},\infty)\):

421:   \(UT(\aleph_0,WO,WO)\): The union of a denumerable set of well orderable sets can be well ordered.

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