Implications

We have the following indirect implication of form equivalence classes:

331 \(\Rightarrow\) 283
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
331 \(\Rightarrow\) 332 Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
332 \(\Rightarrow\) 343 Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
343 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 283 The well-ordered and well-orderable subsets of a set, Truss, J. K. 1973d, Z. Math. Logik Grundlagen Math.

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

Howard-Rubin Number Statement
331:

If \((X_i)_{i\in I}\) is a family of compact non-empty topological spaces then there is a family \((F_i)_{i\in I}\) such that \(\forall i\in I\), \(F_i\) is an irreducible closed subset of \(X_i\).

332:  

A product of non-empty compact sober topological spaces is non-empty.

343:

A product of non-empty, compact \(T_2\) topological spaces is non-empty.

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

283:

Cardinality of well ordered subsets:  For all \(n\in\omega\) and for all infinite \(x\), \(|x^n| < |s(x)|\) where \(s(x)\) is the set of all well orderable subsets of \(x\).

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