Implications

We have the following indirect implication of form equivalence classes:

168 \(\Rightarrow\) 113
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
168 \(\Rightarrow\) 100 clear
100 \(\Rightarrow\) 347 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
347 \(\Rightarrow\) 40 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
40 \(\Rightarrow\) 43 Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc.
On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 \(\Rightarrow\) 113 note-7

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

Howard-Rubin Number Statement
168:

Dual Cantor-Bernstein Theorem:\((\forall x) (\forall y)(|x| \le^*|y|\) and \(|y|\le^* |x|\) implies  \(|x| = |y|)\) .

100:

Weak Partition Principle:  For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\).

347:

Idemmultiple Partition Principle: If \(y\) is idemmultiple (\(2\times y\approx y\)) and \(x\precsim ^* y\), then \(x\precsim y\).

40:

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

43:

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

113:

Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.

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