You're not lying about your belief that there's no problem. You're just not seeing it, and then assuming that you can't see it it's not there. I don't think you're likely to engage in overtly sexist behaviour, and I doubt that most of the people you work with are either. But the problem isn't so much the individual acts of overtly sexist behaviour. Low level sexism is still endemic in society as a whole (advertising may well be less sexist than it was in, say, the 60s, but look at the extent to which gender roles are still used to sell things), and even if you're actively aware of it then you *will* occasionally find yourself interacting with a stereotype of a person rather than the person themselves.
I'm not calling you sexist. I'm not calling any of the people you associate with sexist. I'm saying that you (like everyone, including me) will engage in occasional acts of sexist behaviour, and unless you're able to see that this is what's happening it provides an environment where people feel safe in engaging in overtly sexist acts.