Metamath Proof Explorer


Theorem iooltub

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

Ref Expression
Assertion iooltub ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) → 𝐶 < 𝐵 )

Proof

Step Hyp Ref Expression
1 elioo2 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶𝐶 < 𝐵 ) ) )
2 simp3 ( ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶𝐶 < 𝐵 ) → 𝐶 < 𝐵 )
3 1 2 syl6bi ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) → 𝐶 < 𝐵 ) )
4 3 3impia ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) → 𝐶 < 𝐵 )