We have the following indirect implication of form equivalence classes:

335-n \(\Rightarrow\) 155
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
335-n \(\Rightarrow\) 333 Bases for vector spaces over the two element field and the axiom of choice, Keremedis, K. 1996a, Proc. Amer. Math. Soc.
333 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 106 Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
106 \(\Rightarrow\) 78 Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
78 \(\Rightarrow\) 155 Geordnete Lauchli Kontinuen, Brunner, N. 1983a, Fund. Math.

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

Howard-Rubin Number Statement
335-n:

Every quotient group of an Abelian group each of whose elements has order  \(\le n\) has a set of representatives.

333:

\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd.

67:

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

106:

Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.

78:

Urysohn's Lemma:  If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). Urysohn [1925], pp 290-292.

155:  \(LC\): There are no non-trivial Läuchli continua. (A Läuchli continuum is a strongly connected continuum. Continuum \(\equiv\) compact, connected, Hausdorff space; and strongly connected \(\equiv\) every continuous real valued function is constant.)

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