I was asked to prove every transitive $\in$-totally ordered set is $\in$-well ordered, and I did. However, I didn't use the transitivity, so I thought my proof must be wrong somewhere. Can you help me find the mistake?
Let X be an $\in$-totally ordered set. Let A be a non-empty subset of X. By Axiom of Foundation, there exists an element $a\in A$ such that $a \cap A = \emptyset$. Now let $b\in A$. Since A is totally ordered, a and b are comparable, ie. either $a\in b$, $a=b$ or $b\in a$. We can't have $b\in a$ since it would imply $b \in a \cap A \neq \emptyset$. Then either $a=b$ or $a\in b$. Hence, a is the least element of A. Since every non-empty subset of totally ordered set X has a least element, it is well ordered.