This is Velleman's exercise 3.5.5 (And NO! not a duplicate of Prove that if $A \mathop \triangle B \subseteq A$ then $B\subseteq A$! My question is different):
Prove that if $A \bigtriangleup B\subseteq A$ then $B \subseteq A.$
Since in the definition of a symmetric difference we have disjunction, shouldn't we prove this statement by cases?
So here's my proof of it:
Proof. Let $x$ be an arbitrary element of $B$. Now suppose $x \not\in A$. From $x \in B$ and $x \not\in A$, we get $x \in (B\setminus A)$. We now consider two cases.
Case 1. $x \in (A\setminus B)$. Then by $A \bigtriangleup B \subseteq A$, we have $x \in A$ which is a contradiction.
Case 2. $x \not\in (A\setminus B)$. Since $x \in (B\setminus A)$ and $A \bigtriangleup B \subseteq A$, $x \in A$ which is also a contradiction.
Since by both cases we reached a contradiction then $x \in A$ and since $x$ was arbitrary, $B \subseteq A$.
In other words, in proof by cases (when we have disjunction in the given/hypotheses/premises) when we also use a contradiction, do we need to reach a contradiction for all the cases or just one will be enough?
Thanks in advance.