Metamath Proof Explorer

Theorem lbicc2

Description: The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007) (Revised by FL, 29-May-2014) (Revised by Mario Carneiro, 9-Sep-2015)

Ref Expression
Assertion lbicc2 ( ( 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 elicc1 ⊢ ( ( 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 ) )