Can someone help me understand this proof that a complete graph with 5 vertices is not planar?

Euler's formula is a formula that necessarily applies to planar graphs that says that the number of Vertices minus the number of Edges plus the number of Faces equals 2; V - E + F = 2. Euler's formula is proved elsewhere, and taken as a given here.

By definition, a complete graph with 5 vertices has 5 vertices and 10 edges. Plugging that into Euler's formula tells us it must have 7 faces:

5 - 10 + F = 2

5 + F = 12

F = 7

2E = pF is an equivalency set up by the number of edges and faces and how they relate to each other (each edge has 2 faces and each face has p edges, on average).

A complete graph has no loops (an edge that starts and ends with the same vertex) and therefore cannot have a face with one edge.

A complete graph has no parallel edges (two or more edges that start and end with the same two vertices) and therefore cannot have a face with two edges.

So every face must have 3 or more edges. So p, the average number of edges per face, must be at least 3.

So if 2E = pF and p >= 3 then 2E >= 3F. But we've already established that E = 10 and F = 7 and plugging that in gives us

2\*10 >= 3\*7

20 >= 21