Theorem eliooord

Description: Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008) (Revised by Mario Carneiro, 9-May-2014)

	Ref	Expression
Assertion	eliooord	⊢ ( 𝐴 ∈ ( 𝐵 (,) 𝐶 ) → ( 𝐵 < 𝐴 ∧ 𝐴 < 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		eliooxr	⊢ ( 𝐴 ∈ ( 𝐵 (,) 𝐶 ) → ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) )
2		elioo2	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 ∈ ( 𝐵 (,) 𝐶 ) ↔ ( 𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶 ) ) )
3	1 2	syl	⊢ ( 𝐴 ∈ ( 𝐵 (,) 𝐶 ) → ( 𝐴 ∈ ( 𝐵 (,) 𝐶 ) ↔ ( 𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶 ) ) )
4	3	ibi	⊢ ( 𝐴 ∈ ( 𝐵 (,) 𝐶 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶 ) )
5		3simpc	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶 ) → ( 𝐵 < 𝐴 ∧ 𝐴 < 𝐶 ) )
6	4 5	syl	⊢ ( 𝐴 ∈ ( 𝐵 (,) 𝐶 ) → ( 𝐵 < 𝐴 ∧ 𝐴 < 𝐶 ) )