  We prove that the nonnegative $3$-intermediate Ricci curvature and uniformly
positive $k$-triRic curvature implies rigidity of complete noncompact two-sided
stable minimal hypersurfaces in a Riemannian manifold $(X^5,g)$ with bounded
geometry. The nonnegativity of $3$-intermediate Ricci curvature can be replaced
by nonnegative Ricci and biRic curvature. In particular, there is no complete
noncompact finite index CMC hypersurface in a closed $5$-dimensional manifold
with positive sectional curvature. It extends result of Chodosh-Li-Stryker [to
appear in J. Eur. Math. Soc (2024)] to $5$-dimensions. We also prove that
complete constant mean curvature hypersurfaces in hyperbolic space
$\mathbb{H}^5$ with finite index and the mean curvature greater than
$\frac{\sqrt{65}}{8}$ must be compact. This improves the previous larger bound
$\frac{\sqrt{175}}{\sqrt{148}}$ on the mean curvature.
