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Theorem 5. The set of the finite ordinals is definable by a term of Proof. Note. itself may be defined already in as In other words: Indeed, since it can be proved in that there is more than one set in the universe of sets, denotes the intersection of all the sets (see the discussion above concerning the operator). This intersection is of course empty (a fact that can easily be proved in 6 Conclusion Set abstractions of the form are commonly used in mathematical practice. It is therefore very desirable that a practical formalization of Zermelo-Fraenkel 8 9 [2] includes also a short description of without any details or proofs.
For this reason, in the rest of the paper we shall essentially work with the few metadata introduced above. g. interactive support to forward reasoning, the HELM metadata set also supports indexing of hypotheses. These additional metadata are also useful in all those cases where the conclusion of the statement does not hold enough information to allow a sensible indexing. g. the induction principle for natural numbers, or, more generally, the elimination scheme for an inductive type: the conclusion is a generic universal assertion containing no constant at all.
For we get the usual 3978 candidates with equality in MainConclusion. The case amounts to restrict the attention to the realm of real numbers, reducing the total number of candidates to 195 (just containing equality) plus 515 statements containing R (and possibly other things). On the latter 515 statements we must compute atmost constraint. For we have the following set of constraints: Note that in this case the intersection of the result sets of the two atleast constraints is not empty (it contains 31 elements).