Implications

We have the following indirect implication of form equivalence classes:

50 \(\Rightarrow\) 146
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 \(\Rightarrow\) 49 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
49 \(\Rightarrow\) 30 clear
30 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 146 The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic
note-26

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.

49:

Order Extension Principle: Every partial ordering can be extended to a linear ordering.  Tarski [1924], p 78.

30:

Ordering Principle: Every set can be linearly ordered.

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

146:

\(A(F,A1)\): For every \(T_2\) topological space \((X,T)\), if \(X\) is a continuous finite to one image of an A1 space then \((X,T)\) is  an A1 space. (\((X,T)\) is A1 means if \(U \subseteq  T\) covers \(X\) then \(\exists f : X\rightarrow U\) such that \((\forall x\in X) (x\in f(x)).)\)

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