How do I prove this statement about the cardinalities of finite sets without using the inclusion-exclusion principle? If we were allowed to use the inclusion-exclusion principle:$|A \cup B|=|A|+|B|-|A \cap B|$ then the question statement explicitly states that the two sets are disjoint, so this proof would be trivial.
Let $A$ and $B$ be finite sets.
Assuming for this part that $A$ and $B$ are disjoint, and adopting the recursive definition of cardinality: (the empty set $\varnothing$ is finite with $|\varnothing|=0$. A set $S$ is finite with $|S|=n+1$, if there exists $s \in S$ such that $|S \backslash\{s\}|=n$ for some $n \in \mathbb{N}$. We call $|S|$ the cardinality of $S$), use induction on $|B|$ to show that $A \cup B$ is finite and that
$$
|A \cup B|=|A|+|B| \text {. }
$$

Outline: pick any member $b$ of $B$. Then use the inductive hypothesis on $A \cup (B \setminus \{b\})$, which inductively is finite and has size $|A| + |B \setminus \{b\}|$. Now add in $b$ again.
