Metamath Proof Explorer

Theorem iccleub

Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009)

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

Proof

Step Hyp Ref Expression
1 elicc1 ⊢ 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