Implications

We have the following indirect implication of form equivalence classes:

284 \(\Rightarrow\) 111
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
284 \(\Rightarrow\) 61 note-36
61 \(\Rightarrow\) 250 clear
250 \(\Rightarrow\) 111 clear

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

Howard-Rubin Number Statement
284:

A system of linear equations over a field \(F\) has a solution in \(F\) if and only if every finite sub-system has a solution in \(F\).

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.

111:

\(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set.

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