Axiom Of Choice

We have the following indirect implication of form equivalence classes:

393 \(\Rightarrow\) 401 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
393 \(\Rightarrow\) 121	clear
121 \(\Rightarrow\) 401	clear

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

Howard-Rubin Number	Statement
393:	\(C(LO,WO)\): Every linearly ordered set of non-empty well orderable sets has a choice function.
121:	\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.
401:	\(KW(LO,<\aleph_0)\), The Kinna-Wagner Selection Principle for a linearly ordered set of finite sets: For every linearly ordered set of finite 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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