Theorem lem1

Description: A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Assertion	lem1	$$⊢A\in ℝ\to A-1\le A$$

Proof

Step	Hyp	Ref	Expression
1		ltm1	$$⊢A\in ℝ\to A-1<A$$
2		peano2rem	$$⊢A\in ℝ\to A-1\in ℝ$$
3		ltle	$$⊢\left(A-1\in ℝ\wedge A\in ℝ\right)\to \left(A-1<A\to A-1\le A\right)$$
4	2 3	mpancom	$$⊢A\in ℝ\to \left(A-1<A\to A-1\le A\right)$$
5	1 4	mpd	$$⊢A\in ℝ\to A-1\le A$$