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Definition df-scut 32225
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. (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
StepHypRef Expression
1 cscut 32224 . 2 class |s
2 va . . 3 setvar 𝑎
3 vb . . 3 setvar 𝑏
4 csur 32119 . . . 4 class No
54cpw 4358 . . 3 class 𝒫 No
6 csslt 32222 . . . 4 class <<s
72cv 1636 . . . . 5 class 𝑎
87csn 4377 . . . 4 class {𝑎}
96, 8cima 5321 . . 3 class ( <<s “ {𝑎})
10 vx . . . . . . 7 setvar 𝑥
1110cv 1636 . . . . . 6 class 𝑥
12 cbday 32121 . . . . . 6 class bday
1311, 12cfv 6104 . . . . 5 class ( bday 𝑥)
14 vy . . . . . . . . . . . 12 setvar 𝑦
1514cv 1636 . . . . . . . . . . 11 class 𝑦
1615csn 4377 . . . . . . . . . 10 class {𝑦}
177, 16, 6wbr 4851 . . . . . . . . 9 wff 𝑎 <<s {𝑦}
183cv 1636 . . . . . . . . . 10 class 𝑏
1916, 18, 6wbr 4851 . . . . . . . . 9 wff {𝑦} <<s 𝑏
2017, 19wa 384 . . . . . . . 8 wff (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)
2120, 14, 4crab 3107 . . . . . . 7 class {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}
2212, 21cima 5321 . . . . . 6 class ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
2322cint 4676 . . . . 5 class ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
2413, 23wceq 1637 . . . 4 wff ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
2524, 10, 21crio 6837 . . 3 class (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))
262, 3, 5, 9, 25cmpt2 6879 . 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)})))
271, 26wceq 1637 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)})))
Colors of variables: wff setvar class
This definition is referenced by:  scutval  32237  dmscut  32244  scutf  32245
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