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

Metamath Proof Explorer

Theorem icoub

Description: A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Assertion icoub ( 𝐴 ∈ ℝ* → ¬ 𝐵 ∈ ( 𝐴 [,) 𝐵 ) )

Proof

Step Hyp Ref Expression
1 simpl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐴 ∈ ℝ* )
2 icossxr ⊢ ( 𝐴 [,) 𝐵 ) ⊆ ℝ*
3 id ⊢ ( 𝐵 ∈ ( 𝐴 [,) 𝐵 ) → 𝐵 ∈ ( 𝐴 [,) 𝐵 ) )
4 2 3 sselid ⊢ ( 𝐵 ∈ ( 𝐴 [,) 𝐵 ) → 𝐵 ∈ ℝ* )
5 4 adantl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐵 ∈ ℝ* )
6 simpr ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐵 ∈ ( 𝐴 [,) 𝐵 ) )
7 icoltub ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐵 < 𝐵 )
8 1 5 6 7 syl3anc ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐵 < 𝐵 )
9 xrltnr ⊢ ( 𝐵 ∈ ℝ* → ¬ 𝐵 < 𝐵 )
10 4 9 syl ⊢ ( 𝐵 ∈ ( 𝐴 [,) 𝐵 ) → ¬ 𝐵 < 𝐵 )
11 10 adantl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ( 𝐴 [,) 𝐵 ) ) → ¬ 𝐵 < 𝐵 )
12 8 11 pm2.65da ⊢ ( 𝐴 ∈ ℝ* → ¬ 𝐵 ∈ ( 𝐴 [,) 𝐵 ) )