Implications

We have the following indirect implication of form equivalence classes:

384 \(\Rightarrow\) 241
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
384 \(\Rightarrow\) 14 "Maximal filters, continuity and choice principles", Herrlich, H. 1997, Quaestiones Math.
14 \(\Rightarrow\) 233 Algebraic closures without choice, Banaschewski, B. 1992, Z. Math. Logik Grundlagen Math.
233 \(\Rightarrow\) 242 clear
242 \(\Rightarrow\) 241 clear

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

Howard-Rubin Number Statement
384:

Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter.

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 \({\Bbb Q}\).

241:

Every algebraic closure of \(\Bbb Q\) has a real closed subfield.

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