Metric space non-separability criteria

Let $(X, \rho)$ - a metric-space with topology induced by the metric.

This space is not a separable-space $\iff \exists \varepsilon_0 > 0:,\exists S,(|S| > |\mathbb{N}|) \hookrightarrow \forall x, y \in S: \rho(x, y) \geq \varepsilon_0$.