Prostitution is illegal in Las Vegas as well as the rest of Clark County.
It's a common misconception.
It is illegal in several other counties as well.
First, the NRS (Nevada Revised Statutes) do not allow for prostitution in counties with more than 400,000 residents. That automatically takes Clark County out of the picture, and shortly, Washoe County will be more than 400K population.
Counties with less than 400K population can vote whether they want to license brothels or not. The following counties do not license brothels or permit prostitution (there is a difference):
Ormsby County (ie, Carson City), Washoe County (do not permit prostitution), Douglas County and Eureka County have no licenses issued for brothels.
Eureka County used to have a brothel back in the 70's. Then one night, it "mysteriously" burned down. Shortly thereafter, the county chose to not renew the license and that was that.