Theorem eliood

Description: Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	eliood.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	eliood.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
	eliood.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	eliood.4	⊢ ( 𝜑 → 𝐴 < 𝐶 )
	eliood.5	⊢ ( 𝜑 → 𝐶 < 𝐵 )
Assertion	eliood	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 (,) 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eliood.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		eliood.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		eliood.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		eliood.4	⊢ ( 𝜑 → 𝐴 < 𝐶 )
5		eliood.5	⊢ ( 𝜑 → 𝐶 < 𝐵 )
6		elioo2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
7	1 2 6	syl2anc	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
8	3 4 5 7	mpbir3and	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 (,) 𝐵 ) )