Definition df-scut 27846

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 27845	. 2 class |s
2		va	. . 3 setvar 𝑎
3		vb	. . 3 setvar 𝑏
4		csur 27702	. . . 4 class No
5	4	cpw 4622	. . 3 class 𝒫 No
6		csslt 27843	. . . 4 class <<s
7	2	cv 1536	. . . . 5 class 𝑎
8	7	csn 4648	. . . 4 class {𝑎}
9	6, 8	cima 5703	. . 3 class ( <<s “ {𝑎})
10		vx	. . . . . . 7 setvar 𝑥
11	10	cv 1536	. . . . . 6 class 𝑥
12		cbday 27704	. . . . . 6 class bday
13	11, 12	cfv 6573	. . . . 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 5166	. . . . . . . . 9 wff 𝑎 <<s {𝑦}
18	3	cv 1536	. . . . . . . . . 10 class 𝑏
19	16, 18, 6	wbr 5166	. . . . . . . . 9 wff {𝑦} <<s 𝑏
20	17, 19	wa 395	. . . . . . . 8 wff (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)
21	20, 14, 4	crab 3443	. . . . . . 7 class {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}
22	12, 21	cima 5703	. . . . . 6 class ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
23	22	cint 4970	. . . . 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 7403	. . 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 7450	. 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  27863  dmscut  27874  scutf  27875