Subset Relation is Transitive/Proof 2

Theorem

The subset relation is transitive:

$\paren {R \subseteq S} \land \paren {S \subseteq T} \implies R \subseteq T$

Proof

Let $V$ be a basic universe.

By definition of basic universe, $R$, $S$ and $T$ are all elements of $V$.

By the Axiom of Transitivity, $R$, $S$ and $T$ are all classes.

We are given that $R \subseteq S$ and $S \subseteq T$.

Hence by Subclass of Subclass is Subclass, $R$ is a subclass of $T$.

By Subclass of Set is Set, it follows that $R$ is a subset of $T$.

Hence the result.

$\blacksquare$