This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem iooltubd

Description: An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses iooltubd.1 ⊢ φ → A ∈ ℝ *
iooltubd.2 ⊢ φ → B ∈ ℝ *
iooltubd.3 ⊢ φ → C ∈ A B
Assertion iooltubd ⊢ φ → C < B

Proof

Step Hyp Ref Expression
1 iooltubd.1 ⊢ φ → A ∈ ℝ *
2 iooltubd.2 ⊢ φ → B ∈ ℝ *
3 iooltubd.3 ⊢ φ → C ∈ A B
4 iooltub ⊢ A ∈ ℝ * ∧ B ∈ ℝ * ∧ C ∈ A B → C < B
5 1 2 3 4 syl3anc ⊢ φ → C < B