Subset

A set $A$ is a subset of a set $B$ if all of the elements in $A$ are also in $B$. Note that $A$ may be equal to $B$. This relationship is denoted by $A \subseteq B$.

Definition

$A\subseteq B = \forall x( x\in A \to x\in B )$

Proper subset

A set $A$ is a proper subset of a set $B$ if $A$ is a subset of $B$ but $A\ne B$.

This relationship is denoted by $A \subset B$.