It's not selfish. That's a misnomer. Unless you define your "self" by the collective interests of your individual genes.

Selfish as in I wouldn't want this to happen to me so I won't do it to others.

Self-preservation is more appropriate for what you're describing than selfishness.

Why?

Selfish is meant to describe disregarding others for yourself. Compassion and empathy fly in the face of that, even when it's rooted in self-preservation.

Not if empathy is as I described.

It's more like I have no right to bitch about it if I'm doing it as well. If I set the precedent then hopefully others will be inspired and follow it. It's collectively in everybody's self interest. I know that it won't always pan out, but it definitely won't pan out if nobody is doing it. So I should do it.

And that's also not the biological reason for morality.

Morality is something that exists on a societal level for the maintenance of said society. It was driven genetically mostly by kin selection and reciprocity, in small tribes. Modern civilization seizes on the same basic instincts though it doesn't match up with genetic incentives. That's because modern civilization is a very recent outlier.

If you want to ignore other animals that have shown a sense of morality because you think human's are special, okay.

I think you really need to define what you personally mean by morality too.

Some people tend to think morality is a code set by social standards when I just think what we be believe to be moral tends to be influenced by social standards.

However I believe morality to be about preventing the most harm and suffering as possible and to cause the opposite. Which differs from human to human so often times what societies try to push as 'morality' is actually immoral.