Prove that no matter how 5 edges are removed from the comple

Prove that no matter how 5 edges are removed from the complete graph K_7, the resulting graph is not planar.

Solution

Note that graph is complete hence each vertex is connected by an edge to each of the other 6 vertices. Hence each vertex has 6 edges incident on it.
So removing 5 edges removes at most 5 edges from a single vertex and hence the graph is still connected after removing 5 edges. Graph is clearly finite.
Number of edges after removing 5 edges is, e=(6*7)/2-5=16 edges
Number of vertices, v=7
3v-6=15
Hence, 3v-6<e
But for a finite connected planar graph we must have: 3v-6>=e
Hence resulting graph is not planar