Implications

We have the following indirect implication of form equivalence classes:

378 \(\Rightarrow\) 216
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
378 \(\Rightarrow\) 132 Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.
132 \(\Rightarrow\) 10 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung
10 \(\Rightarrow\) 216

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

Howard-Rubin Number Statement
378:

Restricted Choice for Families of Well Ordered Sets:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that the family of non-empty well orderable subsets of \(Y\) has a choice function.

132:

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

10:

\(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function.

216:

Every infinite tree has either an infinite chain or an infinite antichain.

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