Definition df-ioc 13242

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

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-ioc

Step	Hyp	Ref	Expression
1		cioc 13238	. 2 class (,]
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 11137	. . 3 class ℝ*
5	2	cv 1540	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1540	. . . . . 6 class 𝑧
8		clt 11138	. . . . . 6 class <
9	5, 7, 8	wbr 5089	. . . . 5 wff 𝑥 < 𝑧
10	3	cv 1540	. . . . . 6 class 𝑦
11		cle 11139	. . . . . 6 class ≤
12	7, 10, 11	wbr 5089	. . . . 5 wff 𝑧 ≤ 𝑦
13	9, 12	wa 395	. . . 4 wff (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)
14	13, 6, 4	crab 3393	. . 3 class {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}
15	2, 3, 4, 4, 14	cmpo 7343	. 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
16	1, 15	wceq 1541	1 wff (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})

This definition is referenced by:  iocval  13274  elioc1  13279  iocssxr  13323  iocssicc  13329  iocssioo  13331  ioounsn  13369  snunioc  13372  leordtval2  23120  iocpnfordt  23123  lecldbas  23127  pnfnei  23128  iocmnfcld  24676  xrtgioo  24715  ismbf3d  25575  dvloglem  26577  asindmre  37722  dvasin  37723  iocioodisjd  42332  ioossioc  45511  eliocre  45528  lbioc  45532