Equivalence of Definitions of Sets Separated by Neighborhoods

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Definition 1 implies Definition 2
Let $A, B \subseteq S$ such that:
 * $\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A \cap N_B = \O$

From Subsets of Disjoint Sets are Disjoint then:
 * $U \cap V = \O$

Thus:
 * $\exists U, V \in \tau: A \subseteq U, B \subseteq V: U \cap V = \O$

Definition 2 implies Definition 1
Let $A, B \subseteq S$ such that:
 * $\exists U, V \in \tau: A \subseteq U, B \subseteq V: U \cap V = \O$

Let $N_A = U$ and $N_B = V$.

From Set is Subset of Itself then:
 * $\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A \cap N_B = \O$