Theorem ltle

Description: 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999)

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

Proof

Step	Hyp	Ref	Expression
1		orc	⊢ ( 𝐴 < 𝐵 → ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) )
2		leloe	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )
3	1 2	syl5ibr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )