Is this wrong?
"true but not provable" is easy.
What is an example of a statement that is "provable but not true"?
So you need to assume two contradictory things are true, somewhere deep down in your logical system. As a trivial example, assume that both the following are true:
1. I am a robot. 2. I am not a robot.
And our false statement which we are going to prove is:
Pizza is made of fear.
We know that the following proposition is true:
I am a robot OR pizza is made of fear
Because we know that I am not a robot, the first term in the above must be false, and so the second term must be true. Therefore, pizza is made of fear.
That's really more an example of the principle of explosion, but the above is a tiny logical system which contains statements that are provable, but not true.
In practice, we tend to use logical systems which can't prove all true statements, but don't let you prove false ones. Much more useful, really.