  When $p$ is an odd prime, we prove that the $\mathbb F_p$-cohomology of
$\mathrm{BP}\langle n\rangle$ as a module over the Steenrod algebra determines
the $p$-local spectrum $\mathrm{BP}\langle n\rangle$. In particular, we prove
that the $p$-local spectrum $\mathrm{BP}\langle n\rangle$ only depends on its
$p$-completion $\mathrm{BP}\langle n\rangle_p^\wedge$. As a corollary, this
proves that the $p$-local homotopy type of $\mathrm{BP}\langle n\rangle$ does
not depend on the ideal by which we take the quotient of $\mathrm{BP}$. In the
course of the argument, we show that there is a vanishing line for odd degree
classes in the Adams spectral sequence for endomorphisms of $\mathrm{BP}\langle
n\rangle$. We also prove that there are enough endomorphisms of
$\mathrm{BP}\langle n\rangle$ in a suitable sense. When $p=2$, we obtain the
results for $n\leq 3$.
