Definition df-scut 27632

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 27631	. 2 class |s
2		va	. . 3 setvar 𝑎
3		vb	. . 3 setvar 𝑏
4		csur 27489	. . . 4 class No
5	4	cpw 4594	. . 3 class 𝒫 No
6		csslt 27629	. . . 4 class <<s
7	2	cv 1532	. . . . 5 class 𝑎
8	7	csn 4620	. . . 4 class {𝑎}
9	6, 8	cima 5669	. . 3 class ( <<s “ {𝑎})
10		vx	. . . . . . 7 setvar 𝑥
11	10	cv 1532	. . . . . 6 class 𝑥
12		cbday 27491	. . . . . 6 class bday
13	11, 12	cfv 6533	. . . . 5 class ( bday ‘𝑥)
14		vy	. . . . . . . . . . . 12 setvar 𝑦
15	14	cv 1532	. . . . . . . . . . 11 class 𝑦
16	15	csn 4620	. . . . . . . . . 10 class {𝑦}
17	7, 16, 6	wbr 5138	. . . . . . . . 9 wff 𝑎 <<s {𝑦}
18	3	cv 1532	. . . . . . . . . 10 class 𝑏
19	16, 18, 6	wbr 5138	. . . . . . . . 9 wff {𝑦} <<s 𝑏
20	17, 19	wa 395	. . . . . . . 8 wff (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)
21	20, 14, 4	crab 3424	. . . . . . 7 class {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}
22	12, 21	cima 5669	. . . . . 6 class ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
23	22	cint 4940	. . . . 5 class ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
24	13, 23	wceq 1533	. . . 4 wff ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
25	24, 10, 21	crio 7356	. . 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 7403	. 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 1533	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  27649  dmscut  27660  scutf  27661