Adding Laundry Sink... Help Needed

Hopefully some of you can give me some guidance...

I am hoping to add a laundry sink to my utility/laundry room. Initially, I was going to "T" into my washer drain stack, but was told that isn't possible because of the siphoning of the p-trap and back-flow into the sink. Makes sense now.

So I moved onto the other drain in the room. This drain (see attached picture) was used to drain my AC condensate. There was no visible trap... just an open pipe up through the floor. Then, a couple of years ago we installed a 92% efficiency furnace. They ran a furnace condensate line to the same drain and installed the trap above the floor. Again, see the picture for the current setup. I would like to use this drain for the laundry sink, but would like to continue to use it as a gravity-drain for the furnace and AC condensate. Is this at all possible? Is my only choice to go with a condensate pump and send that waste water elsewhere? Could I wye in under the sink p-trap and delete the trap that is there? Any other thoughts? If the damn furnace condensate line wasn't so low, I would just run it into the sink itself.

Another question... is it possible that there was never a trap installed under the cement for this line? When it was just an open pipe with the AC condensation hose dripping in, we never smelled sewer gas.

Thanks for any thoughts you can throw my way! Very appreciated!

By the way, they tell me that my furnace puts out about 5 gallons of condensate in 24 hours. That number shocks me!

My first concern is that we do not know how or if that drain is connected to the building drain. Since it is next to a water closet, I have a suspicion that it might have been intended for a lavatory drain, but any fixture drains going under the slab are required to be at least 2" pipe. In any case, if you wanted to connect a laundry tray, the drain must be 2". The trap will still require a vent, which will preferably go through the roof. You will need a condensate pump for the furnace as any new fixture drain you install will be too high to gravity drain.

It is my fault for not knowing enough about plumbing... but this has turned into a massive headache. I never thought about venting the laundry sink p-trap. I think maybe the sink will just get returned and I'll learn to live without it.

Are there any "tricks" to find out if that drain has a trap under the slab... without paying to have someone run a camera down or busting out the cement? Could I ensure that it is connected to the building drain by putting my ear to the "main drain" and running some water down this drain?