Prove that if $A ∩ C ⊆ B$ and $a \in C$ then $a \not \in A\setminus B$

Question

Suppose $A ∩ C ⊆ B$ and $a \in C$. Prove that $a \not \in A\setminus B$.

Need to prove that $a \in C \implies a \notin A\setminus B$

$$\tag1 A ∩ C ⊆ B$$ $$\tag2 (a \in A \land a \in C) \implies a \in B$$ $$\tag3 \lnot(a \in A \land a \in C) \lor a \in B$$ $$\tag4 (a \notin A \lor a \notin C) \lor a \in B$$ $$\tag5 a \notin C\lor (a \notin A \lor a \in B)$$ $$\tag6 a \notin C\lor \lnot(a \in A \land a \notin B)$$ $$\tag7 a \in C \implies a \notin A\setminus B$$

$(1) \implies (2) \implies (3) \implies (4) \implies (5) \implies (6) \implies (7) $

Is it accurate?

Answer 1

Intersect $A ∩ C ⊆ B$
on both sides by X\B, where X is that universal set thing to get
$A \setminus B ∩ C = A ∩ C \setminus B ⊆ \emptyset.$
Proof is now immediate.