This paper discusses how integrity consistency constraints between different UML models can be precisely defined at a language level. In doing so, this paper introduces a formal object-oriented metamodeling approach. In the approach, integrity consistency constraints between UML models are defined in terms of invariants of the UML model elements used to define the models at the language-level. Adopting a formal approach, constraints are formally defined using Object-Z. This work demonstrates how integrity consistency constraints for UML models can be precisely defined at the language-level and once completed, the formal description of the consistency constraints will be a precise reference of checking consistency of UML models as well as for tool development.