I'm trying to prove that $X \dot\cup Y \implies X \cap Y = \emptyset$ by way of contraposition and would like to know if the following suffices as a proof:
1$$\lnot(X \cap Y = \emptyset) = X \cap Y \neq \emptyset$$ 2$$\implies \exists x: x \in X \land x \in Y $$ 3$$\implies x \in X \lor x \in Y \neq X \dot\cup Y$$
The two things I'm not totally sure of are firstly the implication from $\land$ to $\lor$ in step no. 3 and secondly if it's necessary to use the existential quantifier. I would also really appreciate it if someone were to tell me how a direct proof would work. Thanks a lot!