Children learn about authority from the way adults handle being mistaken.

Children are studying something other than what you are teaching. What lands is not the rule you stated but what you did when you turned out to be wrong in front of them: whether you went quiet, got louder, blamed the nearest person, or simply said so and moved on. That performance becomes their working idea of what power is allowed to do.

The next time you are wrong in front of a child, say it plainly and without a long apology, then carry on. Being corrected without collapse is one of the more useful things a young person can watch an adult do.