Definition df-right 33290

Description: Define the left options of a surreal. This is the set of surreals that are "closest" on the right to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.)

Assertion

Ref	Expression
df-right	⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))})

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-right

Step	Hyp	Ref	Expression
1		cright 33285	. 2 class R
2		vx	. . 3 setvar 𝑥
3		csur 33149	. . 3 class No
4	2	cv 1536	. . . . . . . 8 class 𝑥
5		vz	. . . . . . . . 9 setvar 𝑧
6	5	cv 1536	. . . . . . . 8 class 𝑧
7		cslt 33150	. . . . . . . 8 class <s
8	4, 6, 7	wbr 5068	. . . . . . 7 wff 𝑥 <s 𝑧
9		vy	. . . . . . . . 9 setvar 𝑦
10	9	cv 1536	. . . . . . . 8 class 𝑦
11	6, 10, 7	wbr 5068	. . . . . . 7 wff 𝑧 <s 𝑦
12	8, 11	wa 398	. . . . . 6 wff (𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦)
13		cbday 33151	. . . . . . . 8 class bday
14	10, 13	cfv 6357	. . . . . . 7 class ( bday ‘𝑦)
15	6, 13	cfv 6357	. . . . . . 7 class ( bday ‘𝑧)
16	14, 15	wcel 2114	. . . . . 6 wff ( bday ‘𝑦) ∈ ( bday ‘𝑧)
17	12, 16	wi 4	. . . . 5 wff ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))
18	17, 5, 3	wral 3140	. . . 4 wff ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))
19	4, 13	cfv 6357	. . . . 5 class ( bday ‘𝑥)
20		cold 33282	. . . . 5 class O
21	19, 20	cfv 6357	. . . 4 class ( O ‘( bday ‘𝑥))
22	18, 9, 21	crab 3144	. . 3 class {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))}
23	2, 3, 22	cmpt 5148	. 2 class (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))})
24	1, 23	wceq 1537	1 wff R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))})

This definition is referenced by: (None)