Implications

We have the following indirect implication of form equivalence classes:

403 \(\Rightarrow\) 357
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
403 \(\Rightarrow\) 324 clear
324 \(\Rightarrow\) 357 clear

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

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

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.)

357:

\(KW(\aleph_0,\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set \(M\) of denumerable sets 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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