Theorem nnz

Description: A positive integer is an integer. (Contributed by NM, 9-May-2004) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022)

	Ref	Expression
Assertion	nnz	|- ( N e. NN -> N e. ZZ )

Proof

Step	Hyp	Ref	Expression
1		nnre	|- ( N e. NN -> N e. RR )
2		3mix2	|- ( N e. NN -> ( N = 0 \/ N e. NN \/ -u N e. NN ) )
3		elz	|- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
4	1 2 3	sylanbrc	|- ( N e. NN -> N e. ZZ )