Implications

We have the following indirect implication of form equivalence classes:

286 \(\Rightarrow\) 65
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
286 \(\Rightarrow\) 65 On the relationship between the Boolean prime ideal theorem and two principles in functional analysis, Bell, J.L. 1971, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.

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.

65:

The Krein-Milman Theorem: Let \(K\) be a compact convex set in a locally convex topological vector space \(X\). Then \(K\) has an extreme point. (An extreme point is a point which is not an interior point of any line segment which lies in  \(K\).) Rubin, H./Rubin, J. [1985] p. 177.

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