Implications

We have the following indirect implication of form equivalence classes:

107 \(\Rightarrow\) 96
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
107 \(\Rightarrow\) 96 Transversal Theory, Mirsky, [1971]

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

Howard-Rubin Number Statement
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 \(|B_{1}| = |B_{2}|\).

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