Can I spread disease from cat to cat?

If I pet a stray, wash my hands thoroughly and then pet my own cat later, can I spread any diseases to my cat? He is vaccinated for fvrcp and rabies but not for fiv.

FIV is only spread with direct cat to cat contact of body fluids - mainly when fighting and mating. FIV wouldn't be spread then. You can potentially spread minor illnesses like a respiratory infection and of course fleas. 

I think it's always a good idea to wash your hands after petting other cats (even if they are not strays) because as stated, there are some diseases that can be transferred that way. But not FIV.

My best friend adopted a cat that she found out was FIV+ after taking him to the vet (he was a stray a coworker had found) and then freaked out about it because we hang out a lot and I have a lot of cats (one of whom was sick - and about to be very sick very soon - at the time). She was afraid of cat-human-cat or cat-human-human-cat contamination. But the vet reassured her it was only direct cat-cat that it could be transferred. For good measure, I also asked my vet about it who said the same thing.