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

Processing math: 100%

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 $X$

$(x, x) \in {\sim}$ and therefore

$x \in \{ y : (x, y) \in {\sim} \}$ .

$\square$