Theorem reltre

Description: For all real numbers there is a smaller real number. (Contributed by AV, 5-Sep-2020)

	Ref	Expression
Assertion	reltre	$$⊢\forall x\in ℝ\exists y\in ℝy<x$$

Proof

Step	Hyp	Ref	Expression
1		peano2rem	$$⊢x\in ℝ\to x-1\in ℝ$$
2		breq1	$$⊢y=x-1\to \left(y<x\leftrightarrow x-1<x\right)$$
3	2	adantl	$$⊢\left(x\in ℝ\wedge y=x-1\right)\to \left(y<x\leftrightarrow x-1<x\right)$$
4		ltm1	$$⊢x\in ℝ\to x-1<x$$
5	1 3 4	rspcedvd	$$⊢x\in ℝ\to \exists y\in ℝy<x$$
6	5	rgen	$$⊢\forall x\in ℝ\exists y\in ℝy<x$$