Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
165 \(\Rightarrow\) 122 clear
122 \(\Rightarrow\) 250 clear
250 \(\Rightarrow\) 111 clear

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

Howard-Rubin Number Statement
165:

\(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function.

122:

\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite 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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