What's the name of this set theory property? $(A \subseteq B) \land (B \subseteq C) \implies (A \subseteq C)$

Question

What's the name of this property in set theory that states that $$(A \subseteq B) \land (B \subseteq C) \implies (A \subseteq C)?$$

Answer 1

This is transitivity. In general, a binary relation $R$ is transitive iff $$x R y\wedge yRz\implies xRz.$$

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Other key properties of binary relations you should know are symmetry (= $xRy\implies yRx$) and reflexivity (= $xRx$). A relation which is transitive, symmetric, and reflexive is called an equivalence relation, and this is a fundamentally important notion. A relation which is transitive and symmetric - like $\subseteq$ - is called a partial order (or sometimes weak partial order - the term "partial order" is often reserved for transitive asymmetric relations).

While not related to your specific question here, I think this additional terminology is worth mentioning here since you'll likely run into it soon as well.