three-point geometry

Three-point geometry is a model for incidence geometry.

The primitive notions of point, line, and incident are given the interpretation

• point: a symbol $A,B,C$
• line: a set of two points $\{A,B\},\{B,C\},\{A,C\}$
• incident: "is an element of"

Is this a model for incidence geometry?

Axiom 1: TRUE (all possible combinations of points can define a line)
Axiom 2: TRUE (all lines are defined by two points)
Axiom 3: TRUE (no line which is incident to all three points)