Axiom Of Choice

We have the following indirect implication of form equivalence classes:

43 \(\Rightarrow\) 77 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
43 \(\Rightarrow\) 77	The Axiom of Choice, Jech, 1973b, page 23

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

Howard-Rubin Number	Statement
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.
77:	A linear ordering of a set \(P\) is a well ordering if and only if \(P\) has no infinite descending sequences. Jech [1973b], p 23.

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