Axiom Of Choice

We have the following indirect implication of form equivalence classes:

112 \(\Rightarrow\) 112 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
112 \(\Rightarrow\) 112	The axiom of choice and linearly ordered sets, Howard, P. 1977, Fund. Math.

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

Howard-Rubin Number	Statement
112:	\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
112:	\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).

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