Definition df-ioc 12819

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 12815	. 2 class (,]
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 10745	. . 3 class ℝ*
5	2	cv 1541	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1541	. . . . . 6 class 𝑧
8		clt 10746	. . . . . 6 class <
9	5, 7, 8	wbr 5027	. . . . 5 wff 𝑥 < 𝑧
10	3	cv 1541	. . . . . 6 class 𝑦
11		cle 10747	. . . . . 6 class ≤
12	7, 10, 11	wbr 5027	. . . . 5 wff 𝑧 ≤ 𝑦
13	9, 12	wa 399	. . . 4 wff (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)
14	13, 6, 4	crab 3057	. . 3 class {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}
15	2, 3, 4, 4, 14	cmpo 7166	. 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  12851  elioc1  12856  iocssxr  12898  iocssicc  12904  iocssioo  12906  ioounsn  12944  snunioc  12947  leordtval2  21956  iocpnfordt  21959  lecldbas  21963  pnfnei  21964  iocmnfcld  23514  xrtgioo  23551  ismbf3d  24399  dvloglem  25383  asindmre  35472  dvasin  35473  ioossioc  42554  eliocre  42571  lbioc  42575