Implications

We have the following indirect implication of form equivalence classes:

39 \(\Rightarrow\) 342-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
39 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 342-n clear

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

Howard-Rubin Number Statement
39:

\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.

8:

\(C(\aleph_{0},\infty)\):

342-n:

(For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\):  Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See Form 166.)

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