Theorem znnsub

Description: The positive difference of unequal integers is a positive integer. (Generalization of nnsub .) (Contributed by NM, 11-May-2004)

	Ref	Expression
Assertion	znnsub	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )

Proof

Step	Hyp	Ref	Expression
1		zre	|- ( M e. ZZ -> M e. RR )
2		zre	|- ( N e. ZZ -> N e. RR )
3		posdif	|- ( ( M e. RR /\ N e. RR ) -> ( M < N <-> 0 < ( N - M ) ) )
4	1 2 3	syl2an	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> 0 < ( N - M ) ) )
5		zsubcl	|- ( ( N e. ZZ /\ M e. ZZ ) -> ( N - M ) e. ZZ )
6	5	ancoms	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N - M ) e. ZZ )
7	6	biantrurd	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 < ( N - M ) <-> ( ( N - M ) e. ZZ /\ 0 < ( N - M ) ) ) )
8	4 7	bitrd	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( ( N - M ) e. ZZ /\ 0 < ( N - M ) ) ) )
9		elnnz	|- ( ( N - M ) e. NN <-> ( ( N - M ) e. ZZ /\ 0 < ( N - M ) ) )
10	8 9	bitr4di	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )