Theorem elioo2

Description: Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007)

	Ref	Expression
Assertion	elioo2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		iooval2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 (,) 𝐵 ) = { 𝑥 ∈ ℝ ∣ ( 𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ) } )
2	1	eleq2d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ 𝐶 ∈ { 𝑥 ∈ ℝ ∣ ( 𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ) } ) )
3		breq2	⊢ ( 𝑥 = 𝐶 → ( 𝐴 < 𝑥 ↔ 𝐴 < 𝐶 ) )
4		breq1	⊢ ( 𝑥 = 𝐶 → ( 𝑥 < 𝐵 ↔ 𝐶 < 𝐵 ) )
5	3 4	anbi12d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ) ↔ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
6	5	elrab	⊢ ( 𝐶 ∈ { 𝑥 ∈ ℝ ∣ ( 𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ) } ↔ ( 𝐶 ∈ ℝ ∧ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
7		3anass	⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
8	6 7	bitr4i	⊢ ( 𝐶 ∈ { 𝑥 ∈ ℝ ∣ ( 𝐴 < 𝑥 ∧ 𝑥 < 𝐵 ) } ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) )
9	2 8	syl6bb	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )