Subset proof, show A⊆B

Question

So I was reviewing this question and Im lost on how to do this question, and Ive seen to of misplaced the notes. The question is as follows: if (A ∩ C) ⊆ (B ∩ C) and (A ∩ C̅) ⊆ (B ∩ C̅)

then A ⊆ B

My attempt so far:

(x∈A ∩ x∈C) ⊆ (x∈B ∩ x∈C)  
(x∈A ∩ x∉C) ⊆ (x∈B ∩ x∉C)

Since x∉C and x∈C => ∅ 

(x∈A ∩ ∅) ⊆ (x∈B ∩ ∅)
(x∈A) ⊆ (x∈B ∩ ∅)
A⊆B

I think this is correct though Im a bit rusty and not sure if this is correct

Answer 1

Take $a\in A$. Then either $a\in C$ or $a\in C^\complement$. If $a\in C$, then $a\in A\cap C$ and therefore $a\in B\cap C$; in particular, $a\in B$. And if $a\in C^\complement$, then $a\in A\cap C^\complement$ and therefore $a\in B\cap C^\complement$; in particular, $a\in B$, again.

Concerning your proof, I don't understand the sentence “Since $x\notin C$ and $x\in C\implies\emptyset$”.

Answer 2

I don't think I understand your attempt. Let me outline a simple proof for you:

Let $x\in A$. Then, we know that either $x\in A\cap C$ or $x\in A\cap\overline{C}$. Using this fact, the hypothesis give us that either $x\in B\cap C$ or $x\in B\cap \overline{C}$. Therefore, $x\in B$ and hence $A\subseteq B$.