Subset Relation is Reflexive
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Theorem
Let $C$ be a class.
The subset relation $\subseteq$ on $C$ is a reflexive relation on $C$.
Proof
\(\ds \forall x \in C: \, \) | \(\ds x\) | \(\subseteq\) | \(\ds x\) | Set is Subset of Itself | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall x \in C: \, \) | \(\ds \tuple {x, x}\) | \(\in\) | \(\ds \subseteq\) |
So $\subseteq$ 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