Metamath Proof Explorer

Theorem iccgelb

Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Thierry Arnoux, 23-Dec-2016)

Ref Expression
Assertion iccgelb ⊢ A ∈ ℝ * ∧ B ∈ ℝ * ∧ C ∈ A B → A ≤ C

Proof

Step Hyp Ref Expression
1 elicc1 ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → C ∈ A B ↔ C ∈ ℝ * ∧ A ≤ C ∧ C ≤ B
2 1 biimpa ⊢ A ∈ ℝ * ∧ B ∈ ℝ * ∧ C ∈ A B → C ∈ ℝ * ∧ A ≤ C ∧ C ≤ B
3 2 simp2d ⊢ A ∈ ℝ * ∧ B ∈ ℝ * ∧ C ∈ A B → A ≤ C
4 3 3impa ⊢ A ∈ ℝ * ∧ B ∈ ℝ * ∧ C ∈ A B → A ≤ C