Equivalence of Definitions of Points Separated by Neighborhoods

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

Proof
Let $x, y \in S$.

From Singleton of Element is Subset then:
 * $x$ and $y$ are separated as points by neighborhoods the singletons $\set x$ and $\set y$ are separated as sets by neighborhoods

From Equivalence of Definitions of Sets Separated by Neighborhoods then:
 * the singletons $\set x$ and $\set y$ are separated as sets by neighborhoods the singletons $\set x$ and $\set y$ are separated as sets by open sets

From Singleton of Element is Subset then:
 * the singletons $\set x$ and $\set y$ are separated as sets by open sets $x$ and $y$ are separated as points by open sets