Implications

We have the following indirect implication of form equivalence classes:

202 \(\Rightarrow\) 403
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
202 \(\Rightarrow\) 398 clear
398 \(\Rightarrow\) 400 clear
400 \(\Rightarrow\) 403 clear

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

Howard-Rubin Number Statement
202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

398:

\(KW(LO,\infty)\), The Kinna-Wagner Selection Principle for a linearly ordered family of sets: For every linearly ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

400:

\(KW(LO,LO)\), The Kinna-Wagner Selection Principle for a linearly ordered set of linearly orderable sets: For every linearly ordered set of linearly orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

403:

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

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