Implications

We have the following indirect implication of form equivalence classes:

359 \(\Rightarrow\) 327
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
359 \(\Rightarrow\) 20 clear
20 \(\Rightarrow\) 121
121 \(\Rightarrow\) 122 clear
122 \(\Rightarrow\) 327 clear

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

Howard-Rubin Number Statement
359:

If \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families  of pairwise disjoint sets and \( |A_{x}| \le |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| \le |\bigcup_{x\in S} B_{x}|\).

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.

122:

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

327:

\(KW(WO,<\aleph_0)\),  The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\)  then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

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