 Projective bundle definition of Stiefel–Whitney classes

Let $p:\mathbb{P}(E)\to X$ and put $u=w_1(L_E)$. The mod-two projective bundle formula makes $H^{\ast }(\mathbb{P}(E);{\mathbb{F}}_2)$ free over $H^{\ast }(X;{\mathbb{F}}_2)$ on $1,u,\dots ,u^{d-1}$. The unique relationdefines the Stiefel–Whitney classes.