Definition:Subclass

Definition

Let $A$ and $B$ be classes.

Then $A$ is a subclass of $B$, and we write $A \subseteq B$, if and only if:

$\forall x: \paren {x \in A \implies x \in B}$

where $x \in A$ denotes that $x$ is an element of $A$.

Proper Subclass

Let $A$ and $B$ be classes.

Let $B$ be a subclass of $A$.

Then $B$ is a proper subclass of $A$ if and only if $B \ne A$.

Also see

• Definition:Subset

• Results about subclasses can be found here.

Sources

• 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Classes
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: Definition $0.1$