Implications

We have the following indirect implication of form equivalence classes:

407 \(\Rightarrow\) 206
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
407 \(\Rightarrow\) 14 Effective equivalents of the Rasiowa-Sikorski lemma, Bacsich, P. D. 1972b, J. London Math. Soc. Ser. 2.
14 \(\Rightarrow\) 63 clear
63 \(\Rightarrow\) 70 clear
70 \(\Rightarrow\) 206 clear

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

Howard-Rubin Number Statement
407:

Let \(B\) be a Boolean algebra, \(b\) a non-zero element of \(B\) and \(\{A_i: i\in\omega\}\) a sequence of subsets of \(B\) such that for each \(i\in\omega\), \(A_i\) has a supremum \(a_i\). Then there exists an ultrafilter \(D\) in \(B\) such that \(b\in D\) and, for each \(i\in\omega\), if \(a_i\in D\), then \(D\cap\ A_i\neq\emptyset\).

14:

BPI: Every Boolean algebra has a prime ideal.

63:

\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
Jech [1973b], p 172 prob 8.5.

70:

There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.

206:

The existence of a non-principal ultrafilter: There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\).

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