Why does a reduction from a P-problem to an NP-complete problem not show that P=NP? Consider the following problem, called BoxDepth: 


  Given a set of $n$ axis-aligned rectangles in the plane, how big is the largest subset of these rectangles that contain a common point?


Say we proved these two statements:


There is a a polynomial-time reduction from BoxDepth to MaxClique.
There is a polynomial-time algorithm for BoxDepth with $O(n^3)$ runtime.


Why don’t these two results imply that P=NP?
Part 1 shows how to solve BoxDepth using MaxClique.

Part 2 shows how to solve BoxDepth directly.

None of the parts says anything about solving MaxClique.