Theorem eluzp1l

Description: Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005)

	Ref	Expression
Assertion	eluzp1l	|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M < N )

Proof

Step	Hyp	Ref	Expression
1		eluzle	|- ( N e. ( ZZ>= ` ( M + 1 ) ) -> ( M + 1 ) <_ N )
2	1	adantl	|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( M + 1 ) <_ N )
3		eluzelz	|- ( N e. ( ZZ>= ` ( M + 1 ) ) -> N e. ZZ )
4		zltp1le	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
5	3 4	sylan2	|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( M < N <-> ( M + 1 ) <_ N ) )
6	2 5	mpbird	|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M < N )