Q: How to prove $A$=$C$-$B$ with given hypotheses? Hypotheses: $A$ ⊆ $C$, $B$ ⊆ $C$, $A$ ∪ $B$ = $C$ and $A$ ∩ $B$ = ∅
Proof:

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*Let $x ∈ A ⇒ x ∈ C ⇔ x ∈ A ∪ B.$
Since $x ∈ A ∧ x ∈ A ∪ B ∧ x ∉ A ∩ B ⇒ x ∉ B.$
Since $x ∈ C ∧ x ∉ B ⇒ x ∈ C - B.$


*Let $x ∈ C - B ⇔ x ∈ C ∧ x ∉ B ⇒ x ∈ C ⇔ x ∈ A ∪ B.$
Since $x ∈ A ∪ B ∧ x ∉ B ⇒ x ∈ A.$
Therefore $A = C - B.$
Is this proof right? if yes, is it abbreviated? should I put more details in it?
