Metamath Proof Explorer

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 syl5ibr ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )