Metamath Proof Explorer

Theorem iooid

Description: An open interval with identical lower and upper bounds is empty. (Contributed by NM, 21-Jun-2007) (Revised by Mario Carneiro, 3-Nov-2013)

Ref Expression
Assertion iooid ( 𝐴 (,) 𝐴 ) = ∅

Proof

Step Hyp Ref Expression
1 xrleid ⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 )
2 1 adantr ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → 𝐴 ≤ 𝐴 )
3 ioo0 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( ( 𝐴 (,) 𝐴 ) = ∅ ↔ 𝐴 ≤ 𝐴 ) )
4 2 3 mpbird ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐴 (,) 𝐴 ) = ∅ )
5 ndmioo ⊢ ( ¬ ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐴 (,) 𝐴 ) = ∅ )
6 4 5 pm2.61i ⊢ ( 𝐴 (,) 𝐴 ) = ∅