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Metamath Proof Explorer

Theorem nnnlt1

Description: A positive integer is not less than one. (Contributed by NM, 18-Jan-2004) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion nnnlt1 ( 𝐴 ∈ ℕ → ¬ 𝐴 < 1 )

Proof

Step Hyp Ref Expression
1 nnge1 ⊢ ( 𝐴 ∈ ℕ → 1 ≤ 𝐴 )
2 1re ⊢ 1 ∈ ℝ
3 nnre ⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ )
4 lenlt ⊢ ( ( 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 1 ≤ 𝐴 ↔ ¬ 𝐴 < 1 ) )
5 2 3 4 sylancr ⊢ ( 𝐴 ∈ ℕ → ( 1 ≤ 𝐴 ↔ ¬ 𝐴 < 1 ) )
6 1 5 mpbid ⊢ ( 𝐴 ∈ ℕ → ¬ 𝐴 < 1 )