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Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnoextendlt 33201 Extending a surreal with a negative sign results in a smaller surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
(𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴)

Theoremnoextendgt 33202 Extending a surreal with a positive sign results in a bigger surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
(𝐴 No 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}))

Theoremnolesgn2o 33203 Given 𝐴 less than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2o, then 𝐵(𝑋) = 2o. (Contributed by Scott Fenton, 6-Dec-2021.)
(((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵𝑋) = 2o)

Theoremnolesgn2ores 33204 Given 𝐴 less than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2o, then (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋). (Contributed by Scott Fenton, 6-Dec-2021.)
(((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))

20.9.22  Surreal Numbers: Ordering

Theoremsltsolem1 33205 Lemma for sltso 33206. The sign expansion relationship totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.)
{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or ({1o, 2o} ∪ {∅})

Theoremsltso 33206 Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.)
<s Or No

20.9.23  Surreal Numbers: Birthday Function

Theorembdayfo 33207 The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
bday : No onto→On

20.9.24  Surreal Numbers: Density

Theoremfvnobday 33208 The value of a surreal at its birthday is . (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.)
(𝐴 No → (𝐴‘( bday 𝐴)) = ∅)

Theoremnosepnelem 33209* Lemma for nosepne 33210. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝐴 No 𝐵 No 𝐴 <s 𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))

Theoremnosepne 33210* The value of two non-equal surreals at the first place they differ is different. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝐴 No 𝐵 No 𝐴𝐵) → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) ≠ (𝐵 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}))

Theoremnosep1o 33211* If the value of a surreal at a separator is 1o then the surreal is lesser. (Contributed by Scott Fenton, 7-Dec-2021.)
(((𝐴 No 𝐵 No 𝐴𝐵) ∧ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) = 1o) → 𝐴 <s 𝐵)

Theoremnosepdmlem 33212* Lemma for nosepdm 33213. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝐴 No 𝐵 No 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))

Theoremnosepdm 33213* The first place two surreals differ is an element of the larger of their domains. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))

Theoremnosepeq 33214* The values of two surreals at a point less than their separators are equal. (Contributed by Scott Fenton, 6-Dec-2021.)
(((𝐴 No 𝐵 No 𝐴𝐵) ∧ 𝑋 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)}) → (𝐴𝑋) = (𝐵𝑋))

Theoremnosepssdm 33215* Given two non-equal surreals, their separator is less than or equal to the domain of one of them. Part of Lemma 2.1.1 of [Lipparini] p. 3. (Contributed by Scott Fenton, 6-Dec-2021.)
((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ⊆ dom 𝐴)

Theoremnodenselem4 33216* Lemma for nodense 33221. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)

Theoremnodenselem5 33217* Lemma for nodense 33221. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 33216 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))

Theoremnodenselem6 33218* The restriction of a surreal to the abstraction from nodenselem4 33216 is still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No )

Theoremnodenselem7 33219* Lemma for nodense 33221. 𝐴 and 𝐵 are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝐶) = (𝐵𝐶)))

Theoremnodenselem8 33220* Lemma for nodense 33221. Give a condition for surreal less than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)
((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 ↔ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))

Theoremnodense 33221* Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD). (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 No (( bday 𝑥) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑥𝑥 <s 𝐵))

20.9.25  Surreal Numbers: Full-Eta Property

Theorembdayimaon 33222 Lemma for full-eta properties. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.)
(𝐴𝑉 → suc ( bday 𝐴) ∈ On)

Theoremnolt02olem 33223 Lemma for nolt02o 33224. If 𝐴(𝑋) is undefined with 𝐴 surreal and 𝑋 ordinal, then dom 𝐴𝑋. (Contributed by Scott Fenton, 6-Dec-2021.)
((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)

Theoremnolt02o 33224 Given 𝐴 less than 𝐵, equal to 𝐵 up to 𝑋, and undefined at 𝑋, then 𝐵(𝑋) = 2o. (Contributed by Scott Fenton, 6-Dec-2021.)
(((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → (𝐵𝑋) = 2o)

Theoremnoresle 33225* Restriction law for surreals. Lemma 2.1.4 of [Lipparini] p. 3. (Contributed by Scott Fenton, 5-Dec-2021.)
(((𝑈 No 𝑆 No ) ∧ (dom 𝑈𝐴 ∧ dom 𝑆𝐴 ∧ ∀𝑔𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)

Theoremnomaxmo 33226* A class of surreals has at most one maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
(𝑆 No → ∃*𝑥𝑆𝑦𝑆 ¬ 𝑥 <s 𝑦)

Theoremnoprefixmo 33227* In any class of surreals, there is at most one value of the prefix property. (Contributed by Scott Fenton, 26-Nov-2021.)
(𝐴 No → ∃*𝑥𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥))

Theoremnosupno 33228* The next several theorems deal with a surreal "supremum". This surreal will ultimately be shown to bound 𝐴 below and bound the restriction of any surreal above. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((𝐴 No 𝐴𝑉) → 𝑆 No )

Theoremnosupdm 33229* The domain of the surreal supremum when there is no maximum. The primary point of this theorem is to change bound variable. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑧 ∣ ∃𝑝𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})

Theoremnosupbday 33230* Birthday bounding law for surreal supremum. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) ⊆ suc ( bday 𝐴))

Theoremnosupfv 33231* The value of surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴𝐺 ∈ dom 𝑈 ∧ ∀𝑣𝐴𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆𝐺) = (𝑈𝐺))

Theoremnosupres 33232* A restriction law for surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴𝐺 ∈ dom 𝑈 ∧ ∀𝑣𝐴𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))

Theoremnosupbnd1lem1 33233* Lemma for nosupbnd1 33239. Establish a soft upper bound. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))

Theoremnosupbnd1lem2 33234* Lemma for nosupbnd1 33239. When there is no maximum, if any member of 𝐴 is a prolongment of 𝑆, then so are all elements of 𝐴 above it. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) = 𝑆)

Theoremnosupbnd1lem3 33235* Lemma for nosupbnd1 33239. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not 2o. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2o)

Theoremnosupbnd1lem4 33236* Lemma for nosupbnd1 33239. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not undefined. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅)

Theoremnosupbnd1lem5 33237* Lemma for nosupbnd1 33239. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not 1o. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 1o)

Theoremnosupbnd1lem6 33238* Lemma for nosupbnd1 33239. Establish a hard upper bound when there is no maximum. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)

Theoremnosupbnd1 33239* Bounding law from below for the surreal supremum. Proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)

Theoremnosupbnd2lem1 33240* Bounding law from above when a set of surreals has a maximum. (Contributed by Scott Fenton, 6-Dec-2021.)
(((𝑈𝐴 ∧ ∀𝑦𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑍 No ) ∧ ∀𝑎𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2o⟩}))

Theoremnosupbnd2 33241* Bounding law from above for the surreal supremum. Proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((𝐴 No 𝐴 ∈ V ∧ 𝑍 No ) → (∀𝑎𝐴 𝑎 <s 𝑍 ↔ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆))

Theoremnoetalem1 33242* Lemma for noeta 33247. Establish that our final surreal really is a surreal. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )

Theoremnoetalem2 33243* Lemma for noeta 33247. 𝑍 is an upper bound for 𝐴. Part of Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, 4-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       (((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑋𝐴) → 𝑋 <s 𝑍)

Theoremnoetalem3 33244* Lemma for noeta 33247. When 𝐴 and 𝐵 are separated, then 𝑍 is a lower bound for 𝐵. Part of Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, 7-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∀𝑏𝐵 𝑍 <s 𝑏)

Theoremnoetalem4 33245* Lemma for noeta 33247. Bound the birthday of 𝑍 above. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))

Theoremnoetalem5 33246* Lemma for noeta 33247. The full statement of the theorem with hypotheses. (Contributed by Scott Fenton, 7-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∃𝑧 No (∀𝑎𝐴 𝑎 <s 𝑧 ∧ ∀𝑏𝐵 𝑧 <s 𝑏 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))

Theoremnoeta 33247* The full-eta axiom for the surreal numbers. This is the single most important property of the surreals. It says that, given two sets of surreals such that one comes completely before the other, there is a surreal lying strictly between the two. Furthermore, there is an upper bound on the birthday of that surreal. Alling's axiom FE. (Contributed by Scott Fenton, 7-Dec-2021.)
(((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐴 𝑥 <s 𝑧 ∧ ∀𝑦𝐵 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))

20.9.26  Surreal numbers - ordering theorems

Syntaxcsle 33248 Declare the syntax for surreal less than or equal.
class ≤s

Definitiondf-sle 33249 Define the surreal less than or equal predicate. Compare df-le 10673. (Contributed by Scott Fenton, 8-Dec-2021.)
≤s = (( No × No ) ∖ <s )

Theoremsltirr 33250 Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → ¬ 𝐴 <s 𝐴)

Theoremslttr 33251 Surreal less than is transitive. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐶) → 𝐴 <s 𝐶))

Theoremsltasym 33252 Surreal less than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴))

Theoremsltlin 33253 Surreal less than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐴 = 𝐵𝐵 <s 𝐴))

Theoremslttrieq2 33254 Trichotomy law for surreal less than. (Contributed by Scott Fenton, 22-Apr-2012.)
((𝐴 No 𝐵 No ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴)))

Theoremslttrine 33255 Trichotomy law for surreals. (Contributed by Scott Fenton, 23-Nov-2021.)
((𝐴 No 𝐵 No ) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))

Theoremslenlt 33256 Surreal less than or equal in terms of less than. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))

Theoremsltnle 33257 Surreal less than in terms of less than or equal. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))

Theoremsleloe 33258 Surreal less than or equal in terms of less than. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵𝐴 = 𝐵)))

Theoremsletri3 33259 Trichotomy law for surreal less than or equal. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No ) → (𝐴 = 𝐵 ↔ (𝐴 ≤s 𝐵𝐵 ≤s 𝐴)))

Theoremsltletr 33260 Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 <s 𝐵𝐵 ≤s 𝐶) → 𝐴 <s 𝐶))

Theoremslelttr 33261 Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 ≤s 𝐵𝐵 <s 𝐶) → 𝐴 <s 𝐶))

Theoremsletr 33262 Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 ≤s 𝐵𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶))

Theoremslttrd 33263 Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝜑𝐴 No )    &   (𝜑𝐵 No )    &   (𝜑𝐶 No )    &   (𝜑𝐴 <s 𝐵)    &   (𝜑𝐵 <s 𝐶)       (𝜑𝐴 <s 𝐶)

Theoremsltletrd 33264 Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝜑𝐴 No )    &   (𝜑𝐵 No )    &   (𝜑𝐶 No )    &   (𝜑𝐴 <s 𝐵)    &   (𝜑𝐵 ≤s 𝐶)       (𝜑𝐴 <s 𝐶)

Theoremslelttrd 33265 Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝜑𝐴 No )    &   (𝜑𝐵 No )    &   (𝜑𝐶 No )    &   (𝜑𝐴 ≤s 𝐵)    &   (𝜑𝐵 <s 𝐶)       (𝜑𝐴 <s 𝐶)

Theoremsletrd 33266 Surreal less than or equal is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝜑𝐴 No )    &   (𝜑𝐵 No )    &   (𝜑𝐶 No )    &   (𝜑𝐴 ≤s 𝐵)    &   (𝜑𝐵 ≤s 𝐶)       (𝜑𝐴 ≤s 𝐶)

20.9.27  Surreal numbers - birthday theorems

Theorembdayfun 33267 The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
Fun bday

Theorembdayfn 33268 The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.)
bday Fn No

Theorembdaydm 33269 The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.)
dom bday = No

Theorembdayrn 33270 The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
ran bday = On

Theorembdayelon 33271 The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by Scott Fenton, 8-Dec-2021.)
( bday 𝐴) ∈ On

Theoremnocvxminlem 33272* Lemma for nocvxmin 33273. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ¬ 𝑋 <s 𝑌))

Theoremnocvxmin 33273* Given a nonempty convex class of surreals, there is a unique birthday-minimal element of that class. (Contributed by Scott Fenton, 30-Jun-2011.)
((𝐴 ≠ ∅ ∧ 𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∃!𝑤𝐴 ( bday 𝑤) = ( bday 𝐴))

Theoremnoprc 33274 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ No ∈ V

20.9.28  Surreal numbers: Conway cuts

Syntaxcsslt 33275 Declare the syntax for surreal set less than.
class <<s

Definitiondf-sslt 33276* Define the relationship that holds iff one set of surreals completely precedes another. (Contributed by Scott Fenton, 7-Dec-2021.)
<<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}

Syntaxcscut 33277 Declare the syntax for the surreal cut operator.
class |s

Definitiondf-scut 33278* 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.)
|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)})))

Theorembrsslt 33279* Binary relation form of the surreal set less-than relation. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))

Theoremssltex1 33280 The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 ∈ V)

Theoremssltex2 33281 The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 ∈ V)

Theoremssltss1 33282 The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 No )

Theoremssltss2 33283 The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 No )

Theoremssltsep 33284* The separation property of surreal set less than. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)

Theoremsssslt1 33285 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)

Theoremsssslt2 33286 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)

Theoremnulsslt 33287 The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Theoremnulssgt 33288 The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No 𝐴 <<s ∅)

Theoremconway 33289* Conway's Simplicity Theorem. Given 𝐴 preceeding 𝐵, there is a unique surreal of minimal length separating them. This is a fundamental property of surreals and will be used (via surreal cuts) to prove many properties later on. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))

Theoremscutval 33290* The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))

Theoremscutcut 33291 Cut properties of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))

Theoremscutbday 33292* The birthday of the surreal cut is equal to the minimum birthday in the gap. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))

Theoremsslttr 33293 Transitive law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐵 <<s 𝐶𝐵 ≠ ∅) → 𝐴 <<s 𝐶)

Theoremssltun1 33294 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)

Theoremssltun2 33295 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))

Theoremscutun12 33296 Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴𝐶) |s (𝐵𝐷)) = (𝐴 |s 𝐵))

Theoremdmscut 33297 The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
dom |s = <<s

Theoremscutf 33298 Functionhood statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
|s : <<s ⟶ No

Theoremetasslt 33299* A restatement of noeta 33247 using set less than. (Contributed by Scott Fenton, 10-Dec-2021.)
(𝐴 <<s 𝐵 → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))

Theoremscutbdaybnd 33300 An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.)
(𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))

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