Implications

We have the following indirect implication of form equivalence classes:

20 \(\Rightarrow\) 33-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
20 \(\Rightarrow\) 121
121 \(\Rightarrow\) 33-n clear

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

Howard-Rubin Number Statement
20:

If \(\{A_{x}: x \in S \}\) and \(\{B_{x}: x \in  S\}\) are families  of pairwise disjoint sets and \( |A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| = |\bigcup_{x\in S} B_{x}|\). Moore [1982] (1.4.12 and 1.7.8).

121:

\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.

33-n:

If \(n\in\omega-\{0,1\}\), \(C(LO,n)\):  Every linearly ordered set of \(n\) element sets has  a choice function.

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