Collapsibility of simplicial complexes of hypergraphs

Let $mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $mathcal{F}subsetmathcal{H}$ with covering number at most $p$ is $left(binom{r+p}{r}-1right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $mathcal{F}subsetmathcal{H}$ is $frac{1}{2}binom{2r}{r}$-collapsible.