Implications

We have the following indirect implication of form equivalence classes:

343 \(\Rightarrow\) 250
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
343 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 250 clear

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

Howard-Rubin Number Statement
343:

A product of non-empty, compact \(T_2\) topological spaces is non-empty.

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

61:

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

250:

\((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function.

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