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 )