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Metamath Proof Explorer

Theorem icoltub

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

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

Proof

Step Hyp Ref Expression
1 elico1 ⊢ 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