Implications

We have the following indirect implication of form equivalence classes:

359 \(\Rightarrow\) 194
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
359 \(\Rightarrow\) 20 clear
20 \(\Rightarrow\) 101 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
101 \(\Rightarrow\) 40 On some weak forms of the axiom of choice in set theory, Pelc, A. 1978, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
40 \(\Rightarrow\) 39 clear
39 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 16 clear
16 \(\Rightarrow\) 194 clear

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

Howard-Rubin Number Statement
359:

If \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families  of pairwise disjoint sets and \( |A_{x}| \le |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| \le |\bigcup_{x\in S} B_{x}|\).

20:

If \(\{A_{x}: x \in S \}\) and \(\{B_{x}: x \in  S\}\) are families  of pairwise disjoint sets and \( |A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| = |\bigcup_{x\in S} B_{x}|\). Moore [1982] (1.4.12 and 1.7.8).

101:

Partition Principle:  If \(S\) is a partition of \(M\), then \(S \precsim M\).

40:

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

39:

\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.

8:

\(C(\aleph_{0},\infty)\):

16:

\(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function.

194:

\(C(\varPi^1_2)\) or \(AC(\varPi^1_2)\): If \(P\in \omega\times{}^{\omega}\omega\), \(P\) has domain \(\omega\), and \(P\) is in \(\varPi^1_2\), then there is a sequence of elements \(\langle x_{k}: k\in\omega\rangle\) of \({}^{\omega}\omega\) with \(\langle k,x_{k}\rangle \in P\) for all \(k\in\omega\). Kanovei [1979].

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