Definition df-scut 27694

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 27693	. 2 class |s
2		va	. . 3 setvar 𝑎
3		vb	. . 3 setvar 𝑏
4		csur 27549	. . . 4 class No
5	4	cpw 4551	. . 3 class 𝒫 No
6		csslt 27691	. . . 4 class <<s
7	2	cv 1539	. . . . 5 class 𝑎
8	7	csn 4577	. . . 4 class {𝑎}
9	6, 8	cima 5622	. . 3 class ( <<s “ {𝑎})
10		vx	. . . . . . 7 setvar 𝑥
11	10	cv 1539	. . . . . 6 class 𝑥
12		cbday 27551	. . . . . 6 class bday
13	11, 12	cfv 6482	. . . . 5 class ( bday ‘𝑥)
14		vy	. . . . . . . . . . . 12 setvar 𝑦
15	14	cv 1539	. . . . . . . . . . 11 class 𝑦
16	15	csn 4577	. . . . . . . . . 10 class {𝑦}
17	7, 16, 6	wbr 5092	. . . . . . . . 9 wff 𝑎 <<s {𝑦}
18	3	cv 1539	. . . . . . . . . 10 class 𝑏
19	16, 18, 6	wbr 5092	. . . . . . . . 9 wff {𝑦} <<s 𝑏
20	17, 19	wa 395	. . . . . . . 8 wff (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)
21	20, 14, 4	crab 3394	. . . . . . 7 class {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}
22	12, 21	cima 5622	. . . . . 6 class ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
23	22	cint 4896	. . . . 5 class ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
24	13, 23	wceq 1540	. . . 4 wff ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
25	24, 10, 21	crio 7305	. . 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 7351	. 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 1540	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  27711  dmscut  27722  scutf  27723