Theorem nn0nlt0

Description: A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	nn0nlt0	⊢ ( 𝐴 ∈ ℕ0 → ¬ 𝐴 < 0 )

Proof

Step	Hyp	Ref	Expression
1		nn0ge0	⊢ ( 𝐴 ∈ ℕ0 → 0 ≤ 𝐴 )
2		0re	⊢ 0 ∈ ℝ
3		nn0re	⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ )
4		lenlt	⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 ≤ 𝐴 ↔ ¬ 𝐴 < 0 ) )
5	2 3 4	sylancr	⊢ ( 𝐴 ∈ ℕ0 → ( 0 ≤ 𝐴 ↔ ¬ 𝐴 < 0 ) )
6	1 5	mpbid	⊢ ( 𝐴 ∈ ℕ0 → ¬ 𝐴 < 0 )