Implications

We have the following indirect implication of form equivalence classes:

345 \(\Rightarrow\) 242
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 \(\Rightarrow\) 233 Algebraic closures without choice, Banaschewski, B. 1992, Z. Math. Logik Grundlagen Math.
233 \(\Rightarrow\) 242 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.

233:

Artin-Schreier theorem: If a field has an algebraic closure it is unique up to isomorphism.

242:

There is, up to an isomorphism, at most one algebraic closure of \({\Bbb Q}\).

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