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 ∈ ( 0 ..^ 𝐴 ) ↔ 𝐴 ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		0z	⊢ 0 ∈ ℤ
2		3anass	⊢ ( ( 0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴 ) ↔ ( 0 ∈ ℤ ∧ ( 𝐴 ∈ ℤ ∧ 0 < 𝐴 ) ) )
3	1 2	mpbiran	⊢ ( ( 0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴 ) ↔ ( 𝐴 ∈ ℤ ∧ 0 < 𝐴 ) )
4		fzolb	⊢ ( 0 ∈ ( 0 ..^ 𝐴 ) ↔ ( 0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 < 𝐴 ) )
5		elnnz	⊢ ( 𝐴 ∈ ℕ ↔ ( 𝐴 ∈ ℤ ∧ 0 < 𝐴 ) )
6	3 4 5	3bitr4i	⊢ ( 0 ∈ ( 0 ..^ 𝐴 ) ↔ 𝐴 ∈ ℕ )