Theorem reltre

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

	Ref	Expression
Assertion	reltre	⊢ ∀ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ 𝑦 < 𝑥

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝑦 = ( 𝑥 − 1 ) → ( 𝑦 < 𝑥 ↔ ( 𝑥 − 1 ) < 𝑥 ) )
2		peano2rem	⊢ ( 𝑥 ∈ ℝ → ( 𝑥 − 1 ) ∈ ℝ )
3		ltm1	⊢ ( 𝑥 ∈ ℝ → ( 𝑥 − 1 ) < 𝑥 )
4	1 2 3	rspcedvdw	⊢ ( 𝑥 ∈ ℝ → ∃ 𝑦 ∈ ℝ 𝑦 < 𝑥 )
5	4	rgen	⊢ ∀ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ 𝑦 < 𝑥