Should I vent my P-trap?

Here is the existing scenario:

My laundry machine drains into a stand pipe, which extends down to a P-trap just under the floor in my crawlspace. Following the trap is a line that runs all the way across the house to the main sewer line. This existing system is not vented, but I have never had any problems with water spilling everywhere or the trap siphoning out. What I have had is problems with the P-trap freezing up because it's in an unheated crawlspace. I've tried insulating the trap, and all it does it make it take longer to unfreeze. I am wary of putting a heat-trace cable on PVC pipe and I don't like leaving a space heater in the crawlspace where I can't get at it.

So - my solution is to re-plumb. Bring the trap above the floor into a heated space, and just dump it into the pipe under the crawlspace all the way back to the sewer... nowhere for it to freeze in the winter. However, I also know that I am supposed to vent a trap.

In my scenario, I cannot vent the trap in the normal fashion... there's no existing vent to the roof nearby to tap into, and I refuse to tear apart my house just to meet this requirement.

So the question becomes: If there was no apparent issue with it being non-vented before, can I just replumb with the trap above the floor, and leave it non-vented?

If I do need to vent it, I'll use an air admittance valve (cheater vent)... I know that they are not the "ideal" solution, but they meet code in my area for doing something like I am doing... a minor renovation that is otherwise impractical to vent appropriately.

Thoughts? I would rather not vent it at all. Then I don't have to trust the cheater vent to stay closed when it's not in use. And like I say, it is not currently vented, and has no issues related to venting whatsoever.

You should to create a vaccume to pull the water out of the trap.