Theorem alrple

Description: Show that A is less than B by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014)

	Ref	Expression
Assertion	alrple	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rexr	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* )
2		xralrple	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + 𝑥 ) ) )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ∀ 𝑥 ∈ ℝ+ 𝐴 ≤ ( 𝐵 + 𝑥 ) ) )