In an incidence geometry, two points are connected by exactly one line, there are (at least) three noncollinear points, and every line contains at least two points. In this context the de Bruijn-Erdos theorem (1948) says that there are at least as many lines as points. There's a proof in the text Combinatorics of Finite Geometries by Lynn Margaret Batten. One can also find proofs on the net by searching for "de Bruijn-Erdos theorem".
George