Definition df-ioc 8762

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 8758	. 2 class (,]
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cxr 7059	. . 3 class ℝ*
5	2	cv 1242	. . . . . 6 class 𝑥
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1242	. . . . . 6 class 𝑧
8		clt 7060	. . . . . 6 class <
9	5, 7, 8	wbr 3764	. . . . 5 wff 𝑥 < 𝑧
10	3	cv 1242	. . . . . 6 class 𝑦
11		cle 7061	. . . . . 6 class ≤
12	7, 10, 11	wbr 3764	. . . . 5 wff 𝑧 ≤ 𝑦
13	9, 12	wa 97	. . . 4 wff (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)
14	13, 6, 4	crab 2310	. . 3 class {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}
15	2, 3, 4, 4, 14	cmpt2 5514	. 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
16	1, 15	wceq 1243	1 wff (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})

This definition is referenced by:  iocval  8787  elioc1  8791  iocssxr  8826  iocssicc  8830  iocssioo  8832