By using the definition, the cardinality is the same iff there is a bijection. So if $B\subset A$ and the exist a injection $f\colon A\rightarrow B$, then $A$ and $B$ has same cardinality.
Elementary Set Theory – Do $A$ and $B$ Have Same Cardinality If $B\subset A$ and There Exists an Injection $A\rightarrow B$?
elementary-set-theory
Best Answer
If $B \subset A$ then there exists an injection $g : B \to A$ defined in the natural way (identity). By existence of $g$ and the injection $f : A \to B,$ invoke Cantor–Bernstein–Schroeder theorem then there exist a bijection $h : A \to B.$ Hence $A$ and $B$ has the same cardinality.