Axiom Of Choice

We have the following indirect implication of form equivalence classes:

67 \(\Rightarrow\) 126 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
67 \(\Rightarrow\) 126	clear

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

Howard-Rubin Number	Statement
67:	\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
126:	\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets 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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