Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
3 \(\Rightarrow\) 9 Cardinal addition and the axiom of choice, Howard, P. 1974, Bull. Amer. Math. Soc.
9 \(\Rightarrow\) 10 Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 216

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

Howard-Rubin Number Statement
3:  \(2m = m\): For all infinite cardinals \(m\), \(2m = m\).
9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

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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