Theorem nngt0

Description: A positive integer is positive. (Contributed by NM, 26-Sep-1999)

	Ref	Expression
Assertion	nngt0	⊢ ( 𝐴 ∈ ℕ → 0 < 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nnre	⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ )
2		nnge1	⊢ ( 𝐴 ∈ ℕ → 1 ≤ 𝐴 )
3		0lt1	⊢ 0 < 1
4		0re	⊢ 0 ∈ ℝ
5		1re	⊢ 1 ∈ ℝ
6		ltletr	⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 0 < 1 ∧ 1 ≤ 𝐴 ) → 0 < 𝐴 ) )
7	4 5 6	mp3an12	⊢ ( 𝐴 ∈ ℝ → ( ( 0 < 1 ∧ 1 ≤ 𝐴 ) → 0 < 𝐴 ) )
8	3 7	mpani	⊢ ( 𝐴 ∈ ℝ → ( 1 ≤ 𝐴 → 0 < 𝐴 ) )
9	1 2 8	sylc	⊢ ( 𝐴 ∈ ℕ → 0 < 𝐴 )