: Edited again think the properties can be devided in the following two groups: 

 Special properties of projective geometry of the Fano 3 space (mentionable) 

 * Each point is contained in 7 lines and 7 planes 
 * Each line is contained in 3 planes and contains 3 points  
 * Each plane contains 7 points and 7 lines 
 * Each plane is isomorphic to the Fano plane 
 * Every pair of distinct planes intersect in a line 
 * A line and a plane intersect in at least one point 

 General properties (no need to mention) 

 * Every pair of distinct points are contained in exactly one line  
 * every trio of noncollinear points is in exactly one plane 
 * every pair of intersecting lines is in exactly one plane   
 * Every pair of distinct planes intersects in exactly one line (important is they do intersect it is always in maximum one line) 
 : I am not sure about "Each plane is isomorphic to the Fano plane" seems to be  a general truth as well and could be removed.