Implications

We have the following indirect implication of form equivalence classes:

168 \(\Rightarrow\) 347
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

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 \(x\precsim y\).

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