Axiom:Axiom of Dependent Choice/Left-Total

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Let $\RR$ be a binary relation on a non-empty set $S$.

Suppose that:

$\forall a \in S: \exists b \in S: a \mathrel \RR b$

that is, that $\RR$ is a left-total relation (specifically a serial relation).

The axiom of dependent choice states that there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $S$ such that:

$\forall n \in \N: x_n \mathrel \RR x_{n + 1}$