New Washing Machine Vent Question

I recently installed a new drain and supply lines for a washing machine (see attached). I tapped into the main waste/soil stack just before it exits the house through the wall. The 18" standpipe is two feet away from the stack. I added a vent to be used with an AAV. My question: do I need the vent? I would prefer not to use the AAV if it's not required. I thought I read somewhere if the appliance is within 5 feet of the main stack, a vent is not necessary.

Any and all help is much appreciated...

The main stack is only considered to be a vent above the point of which the highest fixture drain is connected to it.
The washing machine trap is required to have a vent.

Thanks for your reply, cacher_chick...

The washer should have a 2" p-trap and standpipe.
The vent can be 1.5"