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		peano2rem	⊢ ( 𝑥 ∈ ℝ → ( 𝑥 − 1 ) ∈ ℝ )
2		breq1	⊢ ( 𝑦 = ( 𝑥 − 1 ) → ( 𝑦 < 𝑥 ↔ ( 𝑥 − 1 ) < 𝑥 ) )
3	2	adantl	⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 = ( 𝑥 − 1 ) ) → ( 𝑦 < 𝑥 ↔ ( 𝑥 − 1 ) < 𝑥 ) )
4		ltm1	⊢ ( 𝑥 ∈ ℝ → ( 𝑥 − 1 ) < 𝑥 )
5	1 3 4	rspcedvd	⊢ ( 𝑥 ∈ ℝ → ∃ 𝑦 ∈ ℝ 𝑦 < 𝑥 )
6	5	rgen	⊢ ∀ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ 𝑦 < 𝑥