Stiefel–Whitney classes from the projective-bundle relation

The classes $w_i(E)$ are well defined and have the asserted degrees and conventions. Existence and uniqueness of the coefficients $c_i$ is [F1], so each $w_i(E)$ is a single well-defined element of $H^i(B;{\mathbb{F}}_2)$; the relation is monic because its $x^n$-coefficient is $1$. The conventions $w_0=1$ and $w_i=0$ for $i>n$ extend the definition to all indices, and the total class is the finite sum of the nonzero terms, so it is a class in $H^0\oplus \cdots \oplus H^n$. For the zero bundle no positive coefficients exist and the total class is $1$. The construction consumes AC only through [F1].

The rank-one case. Let $L\to B$ be a numerable real line bundle. By [F2], $P(L)\cong B$ over $B$ and the tautological line ${\gamma }_L$ is $L$, so $x_L\in H^1(P(L);{\mathbb{F}}_2)=H^1(B;{\mathbb{F}}_2)$ and the defining relation of [F1] reads $x_L+w_1(L)=0$ in $H^1(B;{\mathbb{F}}_2)$. Since $-1=1$ in ${\mathbb{F}}_2$, this gives $w_1(L)=x_L$; the conventions give $w_i(L)=0$ for $i\ge 2$ and $w(L)=1+x_L$.

Boundary cases. In rank one the fiber ${\mathbb{R}\mathbb{P}}^0$ is a point, so the base of the relation is the whole base $B$ and the displayed computation is literal. Over the empty base every group is zero, the relation is the zero relation, and the conventions give the zero classes with $w_0=1=0$ in the zero ring. The rank-zero convention $w(0_B)=1$ is the unit, matching the degree-zero convention used in every rank.