Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
380 \(\Rightarrow\) 132 clear
132 \(\Rightarrow\) 73 clear
73 \(\Rightarrow\) 342-n clear

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

Howard-Rubin Number Statement
380:

\(PC(\infty,WO,\infty)\):  For every infinite family of non-empty well orderable sets, there is an infinite subfamily \(Y\) of \(X\) which has a choice function.

132:

\(PC(\infty, <\aleph_0,\infty)\):  Every infinite family of finite  sets has an infinite subfamily with a choice function.

73:

\(\forall n\in\omega\), \(PC(\infty,n,\infty)\):  For every \(n\in\omega\), if \(C\) is an infinite family of \(n\) element sets, then \(C\) has an infinite subfamily with a choice function. De la Cruz/Di Prisco [1998b]

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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