ThmDex – An index of mathematical definitions, results, and conjectures.
Result R716 on D179: Equivalence class
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. $\square$