Definition df-ioc 13311

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 13307	. 2 class (,]
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 11207	. . 3 class ℝ*
5	2	cv 1539	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1539	. . . . . 6 class 𝑧
8		clt 11208	. . . . . 6 class <
9	5, 7, 8	wbr 5107	. . . . 5 wff 𝑥 < 𝑧
10	3	cv 1539	. . . . . 6 class 𝑦
11		cle 11209	. . . . . 6 class ≤
12	7, 10, 11	wbr 5107	. . . . 5 wff 𝑧 ≤ 𝑦
13	9, 12	wa 395	. . . 4 wff (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)
14	13, 6, 4	crab 3405	. . 3 class {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}
15	2, 3, 4, 4, 14	cmpo 7389	. 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
16	1, 15	wceq 1540	1 wff (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})

This definition is referenced by:  iocval  13343  elioc1  13348  iocssxr  13392  iocssicc  13398  iocssioo  13400  ioounsn  13438  snunioc  13441  leordtval2  23099  iocpnfordt  23102  lecldbas  23106  pnfnei  23107  iocmnfcld  24656  xrtgioo  24695  ismbf3d  25555  dvloglem  26557  asindmre  37697  dvasin  37698  iocioodisjd  42308  ioossioc  45490  eliocre  45507  lbioc  45511