Well, that is true, when it comes to these viruses in general. It's a bit too early to tell if you actually become immune or how long you will be immune in this case though.
WHO warned as late as April 24th that antibodies in the blood doesn't necessarily mean that the individual has developed immunity to the infection. We've seen people recovered from Covid-19 and become ill a second time as well. That might've been because they still had the virus the first time though, but it's not certain.
That being said, even if we become immune, it's extremely stupid to risk patients and hospital personnel by letting psychiatric patients go home and not be under lock down when the rest of the hospital is in lock down. If we become sick, we're basically doomed. There's not enough manpower to cover the gaps if we would stay home.
Which is also why they won't test us.