Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
50 \(\Rightarrow\) 14 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
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
50:

Sikorski's  Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141.

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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