If $A$ is a proper subset of $B$ then $|A| < |B|$

Question

Why is this statement false? If $A$ is a proper subset of $B$ then $\forall x \in A, x \in B$ but since $A \ne B$ then $\exists x \in B, x\notin A$

This should mean that $|B| > |A|$, right?

Answer 1

$|A|\le|B|$, and that corresponds to the fact that $A$ injects into $B$.