Implications

We have the following indirect implication of form equivalence classes:

286 \(\Rightarrow\) 131
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
286 \(\Rightarrow\) 40 S´eminaire d’Analyse 1992, Morillon, 1991b,
40 \(\Rightarrow\) 39 clear
39 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 126 clear
126 \(\Rightarrow\) 131 clear

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

Howard-Rubin Number Statement
286:

Extended Krein-Milman Theorem:  Let K be a quasicompact (sometimes called convex-compact), convex subset of a locally convex topological vector space, then K has an extreme point. H. Rubin/J. Rubin [1985], p. 177-178.

40:

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

39:

\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.

8:

\(C(\aleph_{0},\infty)\):

126:

\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).

131:

\(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

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