Of course you can prove any statement. The statements only hold for things which satisfy the axioms. "If X is a model of ZF where 1=0, then X also satisfies a

Of course you can prove any statement. The statements only hold for things which satisfy the axioms. "If X is a model of ZF where 1=0, then X also satisfies anything else you want." (Fine, there is no such X so who cares.) You're getting confused because of the shorthand where statements within the model are stated as if they are statements about the world we live in. That's just a shorthand.