Theorem lbico1

Description: The lower bound belongs to a closed-below, open-above interval. See lbicc2 . (Contributed by FL, 29-May-2014)

	Ref	Expression
Assertion	lbico1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 ∈ ( 𝐴 [,) 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ* )
2		xrleid	⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 )
3	2	3ad2ant1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 ≤ 𝐴 )
4		simp3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 < 𝐵 )
5		elico1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ∈ ( 𝐴 [,) 𝐵 ) ↔ ( 𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝐵 ) ) )
6	5	3adant3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( 𝐴 ∈ ( 𝐴 [,) 𝐵 ) ↔ ( 𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝐵 ) ) )
7	1 3 4 6	mpbir3and	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 ∈ ( 𝐴 [,) 𝐵 ) )