Set Theory: Proving Statements About Sets Let $A, B,$ and $C$ be arbitrary sets taken from the positive integers. 
I have to prove or disprove that: $$ \text{If }A ∩ B ∩ C = ∅, \text{then } (A ⊆ \sim B) \text{ or } (A ⊆ \sim C)$$
Here is my disproof using a counterexample:
If $A = \{ \}$ the empty set,
$B = \{2, 3\}$,
$C = \{4, 5\}$.
With these sets defined for $A, B,$ and $C$, the intersection includes the disjoint set, and then that would lead to $A$ being a subset of $B$ or $A$ being a subset of $C$ which counteracts that 
if $A ∩ B ∩ C = ∅$, then $(A ⊆ \sim B)$ or $(A ⊆ \sim C)$.
Is this a sufficient proof?

As another counterexample, let $A=\mathbb{N}$, and $B$ and $C$ be disjoint sets with at least one element each.
Elaborating: On the one hand, since $B$ and $C$ are disjoint,
$$
A \cap B \cap C = \mathbb{N} \cap B \cap C = B \cap C = \emptyset.
$$
On the other hand, $A = \mathbb{N}$ is not contained in the complement of $B$, since $B$ is not empty, and the same for $C$. 
