Axiom Of Choice

We have the following indirect implication of form equivalence classes:

202 \(\Rightarrow\) 400 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
202 \(\Rightarrow\) 398	clear
398 \(\Rightarrow\) 400	clear

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

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

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