Theorem nnnle0

Description: A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020)

	Ref	Expression
Assertion	nnnle0	⊢ ( 𝐴 ∈ ℕ → ¬ 𝐴 ≤ 0 )

Proof

Step	Hyp	Ref	Expression
1		nngt0	⊢ ( 𝐴 ∈ ℕ → 0 < 𝐴 )
2		0re	⊢ 0 ∈ ℝ
3		nnre	⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ )
4		ltnle	⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐴 ↔ ¬ 𝐴 ≤ 0 ) )
5	4	bicomd	⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ¬ 𝐴 ≤ 0 ↔ 0 < 𝐴 ) )
6	2 3 5	sylancr	⊢ ( 𝐴 ∈ ℕ → ( ¬ 𝐴 ≤ 0 ↔ 0 < 𝐴 ) )
7	1 6	mpbird	⊢ ( 𝐴 ∈ ℕ → ¬ 𝐴 ≤ 0 )