Axiom Of Choice

We have the following indirect implication of form equivalence classes:

214 \(\Rightarrow\) 131 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
214 \(\Rightarrow\) 76	clear
76 \(\Rightarrow\) 131	clear

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

Howard-Rubin Number	Statement
214:	\(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(\|f(y)\|=\aleph_{0}\).
76:	\(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).
131:	\(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

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