Um, no? Unlike with toilet seats, tons of people got it this way. There were even cases of hospital outbreaks due to insufficient sterilization of reusable syringes.
About 270 patients, mostly children, were nosocomially infected with HIV subtype G in seventeen hospitals in 1988–1989. The infection was mainly transmitted between children through contaminated catheters and needles, and from children to mothers through breast-feeding.