Implications

We have the following indirect implication of form equivalence classes:

192 \(\Rightarrow\) 217
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
192 \(\Rightarrow\) 43 Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
43 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 9 Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 217 clear

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

Howard-Rubin Number Statement
192:

\(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\).

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.

8:

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

9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

217:

Every infinite partially ordered set has either an infinite chain or an infinite antichain.

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