Definition df-scut 27837

Description: Define the cut operator on surreal numbers. This operator, which Conway takes as the primitive operator over surreals, picks the surreal lying between two sets of surreals of minimal birthday. Definition from [Gonshor] p. 7. (Contributed by Scott Fenton, 7-Dec-2021.)

Assertion

Ref	Expression
df-scut	⊢ |s = (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))

Distinct variable group:   𝑎,𝑏,𝑥,𝑦

Detailed syntax breakdown of Definition df-scut

Step	Hyp	Ref	Expression
1		cscut 27836	. 2 class |s
2		va	. . 3 setvar 𝑎
3		vb	. . 3 setvar 𝑏
4		csur 27693	. . . 4 class No
5	4	cpw 4622	. . 3 class 𝒫 No
6		csslt 27834	. . . 4 class <<s
7	2	cv 1536	. . . . 5 class 𝑎
8	7	csn 4648	. . . 4 class {𝑎}
9	6, 8	cima 5702	. . 3 class ( <<s “ {𝑎})
10		vx	. . . . . . 7 setvar 𝑥
11	10	cv 1536	. . . . . 6 class 𝑥
12		cbday 27695	. . . . . 6 class bday
13	11, 12	cfv 6572	. . . . 5 class ( bday ‘𝑥)
14		vy	. . . . . . . . . . . 12 setvar 𝑦
15	14	cv 1536	. . . . . . . . . . 11 class 𝑦
16	15	csn 4648	. . . . . . . . . 10 class {𝑦}
17	7, 16, 6	wbr 5169	. . . . . . . . 9 wff 𝑎 <<s {𝑦}
18	3	cv 1536	. . . . . . . . . 10 class 𝑏
19	16, 18, 6	wbr 5169	. . . . . . . . 9 wff {𝑦} <<s 𝑏
20	17, 19	wa 395	. . . . . . . 8 wff (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)
21	20, 14, 4	crab 3438	. . . . . . 7 class {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}
22	12, 21	cima 5702	. . . . . 6 class ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
23	22	cint 4972	. . . . 5 class ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
24	13, 23	wceq 1537	. . . 4 wff ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
25	24, 10, 21	crio 7400	. . 3 class (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))
26	2, 3, 5, 9, 25	cmpo 7447	. 2 class (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))
27	1, 26	wceq 1537	1 wff |s = (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))

This definition is referenced by:  scutval  27854  dmscut  27865  scutf  27866