Definition df-ioo 8915

Description: Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Assertion

Ref	Expression
df-ioo	⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-ioo

Step	Hyp	Ref	Expression
1		cioo 8911	. 2 class (,)
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 7152	. . 3 class ℝ*
5	2	cv 1283	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1283	. . . . . 6 class 𝑧
8		clt 7153	. . . . . 6 class <
9	5, 7, 8	wbr 3785	. . . . 5 wff 𝑥 < 𝑧
10	3	cv 1283	. . . . . 6 class 𝑦
11	7, 10, 8	wbr 3785	. . . . 5 wff 𝑧 < 𝑦
12	9, 11	wa 102	. . . 4 wff (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)
13	12, 6, 4	crab 2352	. . 3 class {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}
14	2, 3, 4, 4, 13	cmpt2 5534	. 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
15	1, 14	wceq 1284	1 wff (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})

This definition is referenced by:  iooex  8930  iooval  8931  elioo3g  8933  elioo1  8934  iooss1  8939  iooss2  8940  eliooxr  8950  iccssioo  8965  ioossicc  8982  ioossico  8985  iocssioo  8986  icossioo  8987  ioossioo  8988  ioof  8994  ioodisj  9015