AITA for punishing my 16-year-old step-daughter after we found out she was bullying a kid for being poor

About 2 months ago my wife (f38) and I (m41) learned my step-daughter(f16) was bullying a girl in school over being poor, getting free lunch at school and not being able to afford neccessties such as her own nice car and stuff.
Our daughter was kinda spoiled, we provided her with everything she needed along with an allowance and a part time job at my company (small family service business). We’ve been considered middle-class, doing things others werent as privileged to do such as buying our daughter a car on her 16th birthday.
I come from a family of immigrants and was considered in poverty growing up, after learning about the bullying i was furious as we thought we didnt raise her to behave that way. She was in honors and top ranking of her class.
I tried to talk to our daughter over why she would do that and i was disturbed to learn it was because she viewed that girl as “trailer trash” which irrated me. The girl from what i learned is very smart and works hard, she bought her own beater car buy herself and works 2 jobs. She considered the money our family had as our families money, so i put her in her place and told her that it was not her money but her mom and I’s money.
I decided from that point i was spoiling my daughter too much, we ended up taking away her latest iphone and replacing it with my old iphone 8 (by switching phones with me) with a talk and text plan . We took away her family credit car,sold her car, along with her macbook and other luxuries.
I also told her should would have to find a job without neopotism and work a minimum wage job like everyone else her age, because i’m done giving her handouts if shes gonna act entitled.
Fast forward 2 months later, she is working at a fast food resturant with us driving her around. She doesnt talk to me unless she needs something like a ride but is very upset with me.
My wife feels like i am taking this too far because its affecting her social status and grades and school I however feel like she needs to be humbled because i cant have a daughter who will disrespect people just because the amount of money they have. I also feel that her behaving this way will affect her younger sister (f12) and how she precieves the world.
AITA for punishing my 16-year-old step-daughter after we found out she was bullying a kid for being poor?