Implications

We have the following indirect implication of form equivalence classes:

392 \(\Rightarrow\) 350
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
392 \(\Rightarrow\) 393 clear
393 \(\Rightarrow\) 165 clear
165 \(\Rightarrow\) 32 clear
32 \(\Rightarrow\) 350 clear

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

Howard-Rubin Number Statement
392:

\(C(LO,LO)\): Every linearly ordered set of linearly orderable sets has a choice function.

393:

\(C(LO,WO)\): Every linearly ordered set of non-empty well orderable sets has a choice function.

165:

\(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function.

32:

\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets  has a choice function.

350:

\(MC(\aleph_0,\aleph_0)\): For every denumerable set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).

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