Wishful thinking! You CANNOT apply to U. of Chicago via non-binding Early Action if you are applying to Yale via the Single-Choice Early Action option.
Yale spells out their Early Action rules pretty clearly on their Web site. They say:
If you are a Single-Choice Early Action applicant to Yale, you may apply to another institution’s early admission program as follows:
You may apply to any college’s non-binding rolling admission program.
You may apply to any public institution at any time provided that admission is non-binding.
You may apply to another college’s Early Decision II program, but only if the notification of admission occurs after January 1. If you are admitted through another college’s Early Decision II binding program, you must withdraw your application from Yale.
You may apply to any institution outside of the United States at any time.
The U. of Chicago’s Early Action option is non-binding but it is NOT a “rolling admission program.”
If you try to skirt the rules and apply to both U. of Chicago via Early Action and also Yale via Single-Choice Early Action, you could get in hot water. If you are offered admission to Yale, you may find that your acceptance is revoked if your Chicago application comes to light.
I realize that all these “rules” can be confusing so I hope that “The Dean” has clarified them for you.