Diagonal Relation is Reflexive (Class Theory)

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Theorem

Let $V$ be a basic universe.

Let $\Delta_V$ denote the diagonal relation on $V$:

$\Delta_V = \set {\tuple {x, x}: x \in V}$


$\Delta_V$ is a reflexive relation.


Proof

\(\ds \forall x \in V: \, \) \(\ds x\) \(=\) \(\ds x\) Definition of Equals
\(\ds \leadsto \ \ \) \(\ds \tuple {x, x}\) \(\in\) \(\ds \Delta_V\) Definition of Diagonal Relation

So $\Delta_V$ is reflexive.

$\blacksquare$


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