Subset Relation is Reflexive

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