Definition df-ioc 13267

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 13263	. 2 class (,]
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 11166	. . 3 class ℝ*
5	2	cv 1541	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1541	. . . . . 6 class 𝑧
8		clt 11167	. . . . . 6 class <
9	5, 7, 8	wbr 5086	. . . . 5 wff 𝑥 < 𝑧
10	3	cv 1541	. . . . . 6 class 𝑦
11		cle 11168	. . . . . 6 class ≤
12	7, 10, 11	wbr 5086	. . . . 5 wff 𝑧 ≤ 𝑦
13	9, 12	wa 395	. . . 4 wff (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)
14	13, 6, 4	crab 3390	. . 3 class {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}
15	2, 3, 4, 4, 14	cmpo 7360	. 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
16	1, 15	wceq 1542	1 wff (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})

This definition is referenced by:  iocval  13299  elioc1  13304  iocssxr  13348  iocssicc  13354  iocssioo  13356  ioounsn  13394  snunioc  13397  leordtval2  23155  iocpnfordt  23158  lecldbas  23162  pnfnei  23163  iocmnfcld  24711  xrtgioo  24750  ismbf3d  25599  dvloglem  26597  asindmre  38015  dvasin  38016  iocioodisjd  42751  ioossioc  45926  eliocre  45943  lbioc  45947