Definition df-ioc 13412

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 13408	. 2 class (,]
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 11323	. . 3 class ℝ*
5	2	cv 1536	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1536	. . . . . 6 class 𝑧
8		clt 11324	. . . . . 6 class <
9	5, 7, 8	wbr 5166	. . . . 5 wff 𝑥 < 𝑧
10	3	cv 1536	. . . . . 6 class 𝑦
11		cle 11325	. . . . . 6 class ≤
12	7, 10, 11	wbr 5166	. . . . 5 wff 𝑧 ≤ 𝑦
13	9, 12	wa 395	. . . 4 wff (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)
14	13, 6, 4	crab 3443	. . 3 class {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}
15	2, 3, 4, 4, 14	cmpo 7450	. 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
16	1, 15	wceq 1537	1 wff (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})

This definition is referenced by:  iocval  13444  elioc1  13449  iocssxr  13491  iocssicc  13497  iocssioo  13499  ioounsn  13537  snunioc  13540  leordtval2  23241  iocpnfordt  23244  lecldbas  23248  pnfnei  23249  iocmnfcld  24810  xrtgioo  24847  ismbf3d  25708  dvloglem  26708  asindmre  37663  dvasin  37664  ioossioc  45410  eliocre  45427  lbioc  45431