Equivalent Characterizations of Finer Equivalence Relation

Theorem
Let $X$ be a set.

Let $\equiv$ and $\sim$ be equivalence relations on $X$.


 * 1) $\equiv$ is finer than $\sim$:
 * $\forall x, y \in X : x \equiv y \implies x \sim y$
 * 1) The graph of $\equiv$ is contained in the graph of $\sim$.
 * 2) Every $\equiv$-equivalence class is contained in a $\sim$-equivalence class.
 * 3) Every $\sim$-equivalence class is saturated under $\equiv$.