Definition df-ioc 13303

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 13299	. 2 class (,]
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 11178	. . 3 class ℝ*
5	2	cv 1541	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1541	. . . . . 6 class 𝑧
8		clt 11179	. . . . . 6 class <
9	5, 7, 8	wbr 5086	. . . . 5 wff 𝑥 < 𝑧
10	3	cv 1541	. . . . . 6 class 𝑦
11		cle 11180	. . . . . 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 7369	. 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  13335  elioc1  13340  iocssxr  13384  iocssicc  13390  iocssioo  13392  ioounsn  13430  snunioc  13433  leordtval2  23177  iocpnfordt  23180  lecldbas  23184  pnfnei  23185  iocmnfcld  24733  xrtgioo  24772  ismbf3d  25621  dvloglem  26612  asindmre  38024  dvasin  38025  iocioodisjd  42752  ioossioc  45922  eliocre  45939  lbioc  45943