Definition df-cuts 27766

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-cuts	⊢ |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-cuts

Step	Hyp	Ref	Expression
1		ccuts 27765	. 2 class |s
2		va	. . 3 setvar 𝑎
3		vb	. . 3 setvar 𝑏
4		csur 27617	. . . 4 class No
5	4	cpw 4542	. . 3 class 𝒫 No
6		cslts 27763	. . . 4 class <<s
7	2	cv 1541	. . . . 5 class 𝑎
8	7	csn 4568	. . . 4 class {𝑎}
9	6, 8	cima 5627	. . 3 class ( <<s “ {𝑎})
10		vx	. . . . . . 7 setvar 𝑥
11	10	cv 1541	. . . . . 6 class 𝑥
12		cbday 27619	. . . . . 6 class bday
13	11, 12	cfv 6492	. . . . 5 class ( bday ‘𝑥)
14		vy	. . . . . . . . . . . 12 setvar 𝑦
15	14	cv 1541	. . . . . . . . . . 11 class 𝑦
16	15	csn 4568	. . . . . . . . . 10 class {𝑦}
17	7, 16, 6	wbr 5086	. . . . . . . . 9 wff 𝑎 <<s {𝑦}
18	3	cv 1541	. . . . . . . . . 10 class 𝑏
19	16, 18, 6	wbr 5086	. . . . . . . . 9 wff {𝑦} <<s 𝑏
20	17, 19	wa 395	. . . . . . . 8 wff (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)
21	20, 14, 4	crab 3390	. . . . . . 7 class {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}
22	12, 21	cima 5627	. . . . . 6 class ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
23	22	cint 4890	. . . . 5 class ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
24	13, 23	wceq 1542	. . . 4 wff ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
25	24, 10, 21	crio 7316	. . 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 7362	. 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 1542	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:  cutsval  27786  dmcuts  27797  cutsf  27798