Metamath Proof Explorer

Theorem iocleub

Description: An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019)

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

Proof

Step Hyp Ref Expression
1 elioc1 ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → C ∈ A B ↔ C ∈ ℝ * ∧ A < C ∧ C ≤ B
2 simp3 ⊢ C ∈ ℝ * ∧ A < C ∧ C ≤ B → C ≤ B
3 1 2 biimtrdi ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → C ∈ A B → C ≤ B
4 3 3impia ⊢ A ∈ ℝ * ∧ B ∈ ℝ * ∧ C ∈ A B → C ≤ B