Theorem uz2m1nn

Description: One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012)

	Ref	Expression
Assertion	uz2m1nn	|- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN )

Proof

Step	Hyp	Ref	Expression
1		eluz2b1	|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) )
2		1z	|- 1 e. ZZ
3		znnsub	|- ( ( 1 e. ZZ /\ N e. ZZ ) -> ( 1 < N <-> ( N - 1 ) e. NN ) )
4	2 3	mpan	|- ( N e. ZZ -> ( 1 < N <-> ( N - 1 ) e. NN ) )
5	4	biimpa	|- ( ( N e. ZZ /\ 1 < N ) -> ( N - 1 ) e. NN )
6	1 5	sylbi	|- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN )