Theorem xrltle

Description: 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006)

	Ref	Expression
Assertion	xrltle	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		orc	⊢ ( 𝐴 < 𝐵 → ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) )
2		xrleloe	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )
3	1 2	imbitrrid	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )