Let $M = (E, S)$ be a matroid and $D$ be the set of all minimal dependent sets of $M$. Prove that if $A_1, A_2 \in D$ such that $A_1 \neq A_2$, and if $x \in A_1 \cup A_2$, then there exists $B \in D$ satisfying $B \subseteq (A_1 \cup A_2) \setminus \{x\}$.
Solution:
Assume there is a $C = A_1 \cup A_2$. Because, $A_1, A_2 \in D$, it follows that $C \in D$. This implies that $B \subset C$, which means that $B$ is independent and therefore we conclude that $B \notin D$.
My proof looks a bit vague and different, and I don't even know whether it is actually true. Any ideas?
Is the question phrased correctly?
See the following link: Preposition 1.2
http://math.mit.edu/~goemans/18438F09/lec9.pdf
This link also contains a proof for preposition 1.2
It is the same axiom, but with intersection in place of union