Is it safe to sell my iPhone with the SIM card installed?

You should have removed it. You cannot delete anything off the SIM, and the iPhone does not write anything to the SIM, however it does contain your wireless account info. But, since it is gone, not a lot you can do about it, but if the card is not active and disabled by the carrier then the person buying the phone will not be able to use it anyway. They will have to get their own and will get rid of that one.

Did you do this:

What to do before selling or giving away your iPhone, iPad, or iPod touch - Apple Support

I realize that you would have gotten a new one. You should have removed it, but it is a moot point now since it is already gone.

If you already sold it, then what difference does it make? It's not like you can hunt it down and remove it.

The SIM card is not a problem. When you activated the new one the old one was automatically deactivated. The MUCH more important question is did you reset the phone to its "new" state, as the link that ckuan posted explains?

As long as you did what is indicated in the support document that ckuan provided, then the buyer should have no problems. And as indicated earlier, your SIM is deactivated, so it could not be used.