 Home Metamath Proof ExplorerTheorem List (p. 319 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

Theorem List for Metamath Proof Explorer - 31801-31900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsltso 31801 Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.)
<s Or No

20.8.23  Surreal Numbers: Birthday Function

Theorembdayfo 31802 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

20.8.24  Surreal Numbers: Density

Theoremfvnobday 31803 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 31804* Lemma for nosepne 31805. (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 31805* 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 31806* 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 31807* Lemma for nosepdm 31808. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝐴 No 𝐵 No 𝐴 <s 𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵))

Theoremnosepdm 31808* 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 31809* 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 31810* 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 31811* Lemma for nodense 31816. 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 31812* Lemma for nodense 31816. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 31811 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 31813* The restriction of a surreal to the abstraction from nodenselem4 31811 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 31814* Lemma for nodense 31816. 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 31815* Lemma for nodense 31816. 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 31816* 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))

20.8.25  Surreal Numbers: Full-Eta Property

Theorembdayimaon 31817 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 31818 Lemma for nolt02o 31819. 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 31819 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 31820* 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 31821* A class of surreals has at most one maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
(𝑆 No → ∃*𝑥𝑆𝑦𝑆 ¬ 𝑥 <s 𝑦)

Theoremnoprefixmo 31822* 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 31823* 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 31824* 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 31825* 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 31826* 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 31827* 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 31828* Lemma for nosupbnd1 31834. 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 31829* Lemma for nosupbnd1 31834. 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 31830* Lemma for nosupbnd1 31834. 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 31831* Lemma for nosupbnd1 31834. 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 31832* Lemma for nosupbnd1 31834. 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 31833* Lemma for nosupbnd1 31834. 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 31834* 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 31835* 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 31836* 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 31837* Lemma for noeta 31842. 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 31838* Lemma for noeta 31842. 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 31839* Lemma for noeta 31842. 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 31840* Lemma for noeta 31842. 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 31841* Lemma for noeta 31842. 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 31842* 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))))

20.8.26  Surreal numbers - ordering theorems

Syntaxcsle 31843 Declare the syntax for surreal less than or equal.
class ≤s

Definitiondf-sle 31844 Define the surreal less than or equal predicate. Compare df-le 10065. (Contributed by Scott Fenton, 8-Dec-2021.)
≤s = (( No × No ) ∖ <s )

Theoremsltirr 31845 Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → ¬ 𝐴 <s 𝐴)

Theoremslttr 31846 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 31847 Surreal less than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴))

Theoremsltlin 31848 Surreal less than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐴 = 𝐵𝐵 <s 𝐴))

Theoremslttrieq2 31849 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 31850 Trichotomy law for surreals. (Contributed by Scott Fenton, 23-Nov-2021.)
((𝐴 No 𝐵 No ) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))

Theoremslenlt 31851 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 31852 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 31853 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 31854 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 31855 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 31856 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 31857 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 31858 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 31859 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 31860 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 31861 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)

20.8.27  Surreal numbers - birthday theorems

Theorembdayfun 31862 The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
Fun bday

Theorembdayfn 31863 The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.)
bday Fn No

Theorembdaydm 31864 The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.)
dom bday = No

Theorembdayrn 31865 The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
ran bday = On

Theorembdayelon 31866 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 31867* Lemma for nocvxmin 31868. 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 31868* 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 31869 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ No ∈ V

20.8.28  Surreal numbers: Conway cuts

Syntaxcsslt 31870 Declare the syntax for surreal set less than.
class <<s

Definitiondf-sslt 31871* 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 31872 Declare the syntax for the surreal cut operator.
class |s

Definitiondf-scut 31873* 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 31874* Binary relationship form of the surreal set less than relationship. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))

Theoremssltex1 31875 The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 ∈ V)

Theoremssltex2 31876 The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 ∈ V)

Theoremssltss1 31877 The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 No )

Theoremssltss2 31878 The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 No )

Theoremssltsep 31879* The separation property of surreal set less than. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)

Theoremsssslt1 31880 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)

Theoremsssslt2 31881 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)

Theoremnulsslt 31882 The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Theoremnulssgt 31883 The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No 𝐴 <<s ∅)

Theoremconway 31884* 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 31885* 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 31886 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 31887* 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 31888 Transitive law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐵 <<s 𝐶𝐵 ≠ ∅) → 𝐴 <<s 𝐶)

Theoremssltun1 31889 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)

Theoremssltun2 31890 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))

Theoremscutun12 31891 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 31892 The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
dom |s = <<s

Theoremscutf 31893 Functionhood statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
|s : <<s ⟶ No

Theoremetasslt 31894* A restatement of noeta 31842 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 31895 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)))

Theoremscutbdaylt 31896 If a surreal lies in a gap and is not equal to the cut, its birthday is greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.)
((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday 𝑋))

Theoremslerec 31897* A comparison law for surreals considered as cuts of sets of surreals. In Conway's treatment, this is the definition of less than or equal. (Contributed by Scott Fenton, 11-Dec-2021.)
(((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑𝐷 𝑋 <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s 𝑌)))

Theoremsltrec 31898* A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)
(((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌)))

20.8.29  Surreal numbers - cuts and options