False. You can do both.
Competition is not a bad thing, and to imply that it is a negative thing for women is ridiculous. This isn't a "DOWN WITH THE PATRIARCHY" post; this is a, healthy competition improves the marketplace, improves your business, betters you. You can compete, AND empower.
I've never heard this quote, or anything similar, directed towards men. It's as if, the only way for women to be celebrated in our society, is for us to continually fill the role of nurturer; for us to follow the rules of perceived femininity. Calling women who are competitive, girls, is not empowering, and directly goes against what this quote is attempting to prove. Assuming that the only way a woman competes with another woman, is by tearing her down, does nothing to help.
There are women athletes who are competitive. Women business owners who are competitive. Women in the government who are competitive. Women that are students who are competitive. WOMEN, who are competitive. Young girls who are competitive, who will grow into competitive women.
To be progressive, one should not use cliche quotes, that generalize women in a category that is thinly veiled in misogyny