Metamath Proof Explorer

Theorem ioogtlb

Description: An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion ioogtlb ⊢ A ∈ ℝ * ∧ B ∈ ℝ * ∧ C ∈ A B → A < C

Proof

Step Hyp Ref Expression
1 elioo2 ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → C ∈ A B ↔ C ∈ ℝ ∧ A < C ∧ C < B
2 simp2 ⊢ C ∈ ℝ ∧ A < C ∧ C < B → A < C
3 1 2 biimtrdi ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → C ∈ A B → A < C
4 3 3impia ⊢ A ∈ ℝ * ∧ B ∈ ℝ * ∧ C ∈ A B → A < C