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Definition df-scut 32236
 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 32235 . 2 class |s
2 va . . 3 setvar 𝑎
3 vb . . 3 setvar 𝑏
4 csur 32130 . . . 4 class No
54cpw 4297 . . 3 class 𝒫 No
6 csslt 32233 . . . 4 class <<s
72cv 1630 . . . . 5 class 𝑎
87csn 4316 . . . 4 class {𝑎}
96, 8cima 5252 . . 3 class ( <<s “ {𝑎})
10 vx . . . . . . 7 setvar 𝑥
1110cv 1630 . . . . . 6 class 𝑥
12 cbday 32132 . . . . . 6 class bday
1311, 12cfv 6031 . . . . 5 class ( bday 𝑥)
14 vy . . . . . . . . . . . 12 setvar 𝑦
1514cv 1630 . . . . . . . . . . 11 class 𝑦
1615csn 4316 . . . . . . . . . 10 class {𝑦}
177, 16, 6wbr 4786 . . . . . . . . 9 wff 𝑎 <<s {𝑦}
183cv 1630 . . . . . . . . . 10 class 𝑏
1916, 18, 6wbr 4786 . . . . . . . . 9 wff {𝑦} <<s 𝑏
2017, 19wa 382 . . . . . . . 8 wff (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)
2120, 14, 4crab 3065 . . . . . . 7 class {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}
2212, 21cima 5252 . . . . . 6 class ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
2322cint 4611 . . . . 5 class ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
2413, 23wceq 1631 . . . 4 wff ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})
2524, 10, 21crio 6753 . . 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 6795 . 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 1631 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  32248  dmscut  32255  scutf  32256
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