ThmDex – An index of mathematical definitions, results, and conjectures.
Result R714 on D179: Equivalence class
Element belongs to its own equivalence class
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} x \in \{ y : (x, y) \in {\sim} \} \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 $X$
By definition, an D178: Equivalence relation is a D287: Reflexive binary relation. Thus, $(x, x) \in {\sim}$ and therefore $x \in \{ y : (x, y) \in {\sim} \}$. $\square$