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	$$⊢A\in \mathbb{N}\to \left(A\le 1\leftrightarrow A=1\right)$$

Proof

Step	Hyp	Ref	Expression
1		nnge1	$$⊢A\in \mathbb{N}\to 1\le A$$
2	1	biantrud	$$⊢A\in \mathbb{N}\to \left(A\le 1\leftrightarrow \left(A\le 1\wedge 1\le A\right)\right)$$
3		nnre	$$⊢A\in \mathbb{N}\to A\in ℝ$$
4		1re	$$⊢1\in ℝ$$
5		letri3	$$⊢\left(A\in ℝ\wedge 1\in ℝ\right)\to \left(A=1\leftrightarrow \left(A\le 1\wedge 1\le A\right)\right)$$
6	3 4 5	sylancl	$$⊢A\in \mathbb{N}\to \left(A=1\leftrightarrow \left(A\le 1\wedge 1\le A\right)\right)$$
7	2 6	bitr4d	$$⊢A\in \mathbb{N}\to \left(A\le 1\leftrightarrow A=1\right)$$