Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
295 \(\Rightarrow\) 30 "Dense orderings, partitions, and weak forms of choice", Gonzalez, C. 1995a, Fund. Math.
30 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 216

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

Howard-Rubin Number Statement
295:

DO:  Every infinite set has a dense linear ordering.

30:

Ordering Principle: Every set can be linearly ordered.

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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