The question from Velleman's How to Prove It section 3.1 is as follows:
Suppose $ A \setminus B \subseteq C \cap D $ and $ x \in A. $ Prove that if $ x \notin D$ then $x \in B$
My logical form of the statement is:
$x \in A \land x \in B \rightarrow x \in C \land x \in D$ and $ x \subseteq A$
I am unsure of how to properly express this as a proof. There is this one.
But that thread doesn't show a written out proof.
Also, are you allowed to do the contrapositive of a given? Because if I am, then I could apply demorgan's law to it, which would show that $x \notin D \lor x \notin C \rightarrow x \notin A \lor x \notin B$. Obviously from here, we assume that $ x \notin D$, and from the contrapositive we see that x doesn't belong in B.
For my thinking above, how would I turn this into a proof if correct?