Implications

We have the following indirect implication of form equivalence classes:

393 \(\Rightarrow\) 5
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
393 \(\Rightarrow\) 165 clear
165 \(\Rightarrow\) 32 clear
32 \(\Rightarrow\) 5 clear

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

Howard-Rubin Number Statement
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.

5:

\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.

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