Axiom Of Choice

We have the following indirect implication of form equivalence classes:

394 \(\Rightarrow\) 324 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
394 \(\Rightarrow\) 165	clear
165 \(\Rightarrow\) 324	clear

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

Howard-Rubin Number	Statement
394:	\(C(WO,LO)\): Every well ordered set of non-empty linearly orderable sets has a choice function.
165:	\(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function.
324:	\(KW(WO,WO)\), The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set \(M\) of well orderable sets, there is a function \(f\) such that for all \(A\in M\), if \(\|A\| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

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