Implications

We have the following indirect implication of form equivalence classes:

398 \(\Rightarrow\) 250
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
398 \(\Rightarrow\) 322 clear
322 \(\Rightarrow\) 324 clear
324 \(\Rightarrow\) 327 clear
327 \(\Rightarrow\) 250 clear

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

Howard-Rubin Number Statement
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 \(\emptyset\neq f(A)\subsetneq A\).

322:

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

324:

\(KW(WO,WO)\), The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set \(M\) of well orderable 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.)

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.)

250:

\((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function.

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