Theorem nnle1eq1

Description: A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005)

	Ref	Expression
Assertion	nnle1eq1	⊢ ( 𝐴 ∈ ℕ → ( 𝐴 ≤ 1 ↔ 𝐴 = 1 ) )

Proof

Step	Hyp	Ref	Expression
1		nnge1	⊢ ( 𝐴 ∈ ℕ → 1 ≤ 𝐴 )
2	1	biantrud	⊢ ( 𝐴 ∈ ℕ → ( 𝐴 ≤ 1 ↔ ( 𝐴 ≤ 1 ∧ 1 ≤ 𝐴 ) ) )
3		nnre	⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ )
4		1re	⊢ 1 ∈ ℝ
5		letri3	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝐴 = 1 ↔ ( 𝐴 ≤ 1 ∧ 1 ≤ 𝐴 ) ) )
6	3 4 5	sylancl	⊢ ( 𝐴 ∈ ℕ → ( 𝐴 = 1 ↔ ( 𝐴 ≤ 1 ∧ 1 ≤ 𝐴 ) ) )
7	2 6	bitr4d	⊢ ( 𝐴 ∈ ℕ → ( 𝐴 ≤ 1 ↔ 𝐴 = 1 ) )