Theorem lbfzo0

Description: An integer is strictly greater than zero iff it is a member of NN . (Contributed by Mario Carneiro, 29-Sep-2015)

	Ref	Expression
Assertion	lbfzo0	|- ( 0 e. ( 0 ..^ A ) <-> A e. NN )

Proof

Step	Hyp	Ref	Expression
1		0z	|- 0 e. ZZ
2		3anass	|- ( ( 0 e. ZZ /\ A e. ZZ /\ 0 < A ) <-> ( 0 e. ZZ /\ ( A e. ZZ /\ 0 < A ) ) )
3	1 2	mpbiran	|- ( ( 0 e. ZZ /\ A e. ZZ /\ 0 < A ) <-> ( A e. ZZ /\ 0 < A ) )
4		fzolb	|- ( 0 e. ( 0 ..^ A ) <-> ( 0 e. ZZ /\ A e. ZZ /\ 0 < A ) )
5		elnnz	|- ( A e. NN <-> ( A e. ZZ /\ 0 < A ) )
6	3 4 5	3bitr4i	|- ( 0 e. ( 0 ..^ A ) <-> A e. NN )