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.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering