  In this note, we give a new and short proof for a theorem of Bodineau stating
that the slab percolation threshold $\hat{p}_c$ for the FK-Ising model
coincides with the standard percolation critical point $p_c$ in all dimensions
$d\geq3$. Both proofs rely on the positivity of the surface tension for $p>p_c$
proved by Lebowitz & Pfister. The key difference is that while Bodineau's proof
is based on a delicate dynamic renormalization inspired by the work of Barsky,
Grimmett & Newman, our proof utilizes a technique of Benjamini & Tassion to
prove the uniqueness of macroscopic clusters via sprinkling, which then implies
percolation on slabs through a rather straightforward static renormalization.
