I was just curious for finite sets $A,B$ if $A \subseteq B$ and $|A|=|B|$ then $A=B$. I attempted the proof however I feel I'm missing something.
Proof. To show $A=B$ it remains to show that $B\subseteq A$. Consider the set $B\setminus A$. Since $|A|=|B|$ and $A\subseteq B$, if follows that $B\setminus A$ is empty (How do I argue that this is true?). Hence, $B\subseteq A$. Finally, $A=B$.
Is my proof correct? What are some alternative ways?
Claim: If $A$ and $B$ are finite sets such that $A \subseteq B$ and $\lvert A \rvert = \lvert B \rvert$, then $A = B$.
Proof: Since $A \subseteq B$, there is a natural injection $\iota \colon A \to B$ defined by $\iota(x) = x$ for every $x \in A$. It is well known that an injection between finite sets of equal cardinality must be a bijection. Therefore, $\iota$ is a bijection from $A$ to $B$ that is nothing but the identity function. Hence, $A = B$.