Theorem ubioc1

Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 . (Contributed by FL, 29-May-2014)

	Ref	Expression
Assertion	ubioc1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ( 𝐴 (,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simp2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ* )
2		simp3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 < 𝐵 )
3		xrleid	⊢ ( 𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵 )
4	3	3ad2ant2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ≤ 𝐵 )
5		elioc1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) )
6	5	3adant3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( 𝐵 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) )
7	1 2 4 6	mpbir3and	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ( 𝐴 (,] 𝐵 ) )