Implications

We have the following indirect implication of form equivalence classes:

345 \(\Rightarrow\) 235
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
345 \(\Rightarrow\) 14 Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
14 \(\Rightarrow\) 107
107 \(\Rightarrow\) 96 Transversal Theory, Mirsky, [1971]
96 \(\Rightarrow\) 235 clear

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

Howard-Rubin Number Statement
345:

Rasiowa-Sikorski Axiom:  If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\).

14:

BPI: Every Boolean algebra has a prime ideal.

107:  

M. Hall's Theorem: Let \(\{S(\alpha): \alpha\in A\}\) be a collection of finite subsets (of a set \(X\)) then if

(*) for each finite \(F \subseteq  A\) there is an injective choice function on \(F\)
then there is an injective choice function on \(A\). (That is, a 1-1 function \(f\) such that \((\forall\alpha\in A)(f(\alpha)\in S(\alpha))\).) (According to a theorem of P. Hall (\(*\)) is equivalent to \(\left |\bigcup_{\alpha\in F} S(\alpha)\right|\ge |F|\). P. Hall's theorem does not require the axiom of choice.)

96:

Löwig's Theorem:If \(B_{1}\) and \(B_{2}\) are both bases for the vector space \(V\) then \(|B_{1}| = |B_{2}|\).

235:

If \(V\) is a vector space and \(B_{1}\) and \(B_{2}\) are bases for \(V\) then \(|B_{1}|\) and \(|B_{2}|\) are comparable.

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