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?