ThmDex – An index of mathematical definitions, results, and conjectures.

Equivalence class is not empty

Formulation 1

Let $X$ be a D11: Set such that

(i)	\begin{equation} X \neq \emptyset \end{equation}
(ii)	$x \in X$ is a D2218: Set element in $X$
(iii)	${\sim} \subseteq X \times X$ is an D178: Equivalence relation on $X$

Then \begin{equation} \{ y : (x, y) \in {\sim} \} \neq \emptyset \end{equation}

Proofs

Proof 0

Let $X$ be a D11: Set such that

(i)	\begin{equation} X \neq \emptyset \end{equation}
(ii)	$x \in X$ is a D2218: Set element in $X$
(iii)	${\sim} \subseteq X \times X$ is an D178: Equivalence relation on $X$

Result R714: Element belongs to its own equivalence class shows that $x \in \{ y : (x, y) \in {\sim} \}$, which guarantees that the equivalence class $\{ y : (x, y) \in {\sim} \}$ is not empty. $\square$