Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
107 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 61 clear
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
107:  

M. Hall's Theorem: Let \(\{S(\alpha): \alpha\in A\}\) be a collection of finite subsets (of a set \(X\)) then if

(*) for each finite \(F \subseteq  A\) there is an injective choice function on \(F\)
then there is an injective choice function on \(A\). (That is, a 1-1 function \(f\) such that \((\forall\alpha\in A)(f(\alpha)\in S(\alpha))\).) (According to a theorem of P. Hall (\(*\)) is equivalent to \(\left |\bigcup_{\alpha\in F} S(\alpha)\right|\ge |F|\). P. Hall's theorem does not require the axiom of choice.)

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.

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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