Definition df-ioo 9516

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 9512	. 2 class (,)
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 7671	. . 3 class ℝ*
5	2	cv 1298	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1298	. . . . . 6 class 𝑧
8		clt 7672	. . . . . 6 class <
9	5, 7, 8	wbr 3875	. . . . 5 wff 𝑥 < 𝑧
10	3	cv 1298	. . . . . 6 class 𝑦
11	7, 10, 8	wbr 3875	. . . . 5 wff 𝑧 < 𝑦
12	9, 11	wa 103	. . . 4 wff (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)
13	12, 6, 4	crab 2379	. . 3 class {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}
14	2, 3, 4, 4, 13	cmpo 5708	. 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
15	1, 14	wceq 1299	1 wff (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})

This definition is referenced by:  iooex  9531  iooval  9532  elioo3g  9534  elioo1  9535  iooss1  9540  iooss2  9541  eliooxr  9551  iccssioo  9566  ioossicc  9583  ioossico  9586  iocssioo  9587  icossioo  9588  ioossioo  9589  ioof  9595  ioodisj  9617