The definition of a valid 3D parcel must be correct and unambiguous, because an error or ambiguity in the definition of the extent of a property can lead to expensive legal disputes or to problems with handling 3D parcels in the information systems or problems during data transfer between two systems. This paper develops a rigorous axiomatic definition of a 3D parcel, and its relationship with adjoining parcels within a space partition. Since the requirements of different jurisdictions mandate different levels of validation, some of the axioms are identified as optional. For example, a jurisdiction may require that a parcel must be contiguous, while another may not require this. In earlier publications the axioms concerning valid 3D parcels (within a partition) are formulated in natural language. In this paper we will further formalize this by using mathematical expressions. We also want to prove the necessity of all axioms, i.e. is our set of axioms minimal or are they perhaps overlapping? We show that one of the earlier proposed axioms (A4) is implied by axiom A5 (see discussion in section 3.3) and can be omitted. In order to demonstrate the necessity and independence of the remaining set of axioms, a series of test cases is presented. Each case violates a single axiom and passes all other axioms, thus showing that the set of axioms is non-redundant. In addition, real examples of 3D parcels (From Queensland, Australia), are tested against the validation suite.