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Metamath Proof Explorer

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 )