Theorem rpltrp

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

	Ref	Expression
Assertion	rpltrp	⊢ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ+ 𝑦 < 𝑥

Proof

Step	Hyp	Ref	Expression
1		rphalfcl	⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 / 2 ) ∈ ℝ+ )
2		breq1	⊢ ( 𝑦 = ( 𝑥 / 2 ) → ( 𝑦 < 𝑥 ↔ ( 𝑥 / 2 ) < 𝑥 ) )
3	2	adantl	⊢ ( ( 𝑥 ∈ ℝ+ ∧ 𝑦 = ( 𝑥 / 2 ) ) → ( 𝑦 < 𝑥 ↔ ( 𝑥 / 2 ) < 𝑥 ) )
4		rphalflt	⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 / 2 ) < 𝑥 )
5	1 3 4	rspcedvd	⊢ ( 𝑥 ∈ ℝ+ → ∃ 𝑦 ∈ ℝ+ 𝑦 < 𝑥 )
6	5	rgen	⊢ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ+ 𝑦 < 𝑥