We have the following indirect implication of form equivalence classes:

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

Implication Reference
333 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 112 clear

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

Howard-Rubin Number Statement
333:

\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd.

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).

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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