P. 280 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	leltstr 27901	Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶))
Theorem	lestr 27902	Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶))
Theorem	ltstrd 27903	Surreal less-than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → 𝐵 <s 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 <s 𝐶)
Theorem	ltlestrd 27904	Surreal less-than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → 𝐵 ≤s 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 <s 𝐶)
Theorem	leltstrd 27905	Surreal less-than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐴 ≤s 𝐵)    &   ⊢ (𝜑 → 𝐵 <s 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 <s 𝐶)
Theorem	lestrd 27906	Surreal less-than or equal is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐴 ≤s 𝐵)    &   ⊢ (𝜑 → 𝐵 ≤s 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≤s 𝐶)
Theorem	lesid 27907	Surreal less-than or equal is reflexive. Theorem 0(iii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴)
Theorem	lestric 27908	Surreal trichotomy law. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ∨ 𝐵 ≤s 𝐴))
Theorem	maxs1 27909	A surreal is less than or equal to the maximum of it and another. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ (𝐴 ∈ No → 𝐴 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))
Theorem	maxs2 27910	A surreal is less than or equal to the maximum of it and another. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))
Theorem	mins1 27911	The minimum of two surreals is less than or equal to the first. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴)
Theorem	mins2 27912	The minimum of two surreals is less than or equal to the second. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ (𝐵 ∈ No → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐵)
Theorem	ltlesd 27913	Surreal less-than implies less-than or equal. (Contributed by Scott Fenton, 16-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≤s 𝐵)
Theorem	ltsne 27914	Surreal less-than implies not equal. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴)
Theorem	ltlesnd 27915	Surreal less-than in terms of less-than or equal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≠ 𝐴)))
15.2.2  Birthday Theorems
Theorem	bdayfun 27916	The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ Fun bday
Theorem	bdayfn 27917	The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.)
⊢ bday Fn No
Theorem	bdaydm 27918	The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ dom bday = No
Theorem	bdaydmOLD 27919	Obsolete version of bdaydm 27918 as of 10-Jun-2026. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ dom bday = No
Theorem	bdayrn 27920	The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ ran bday = On
Theorem	bdayon 27921	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
Theorem	nobdaymin 27922*	Any non-empty class of surreals has a birthday-minimal element. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ( bday ‘𝑥) = ∩ ( bday “ 𝐴))
Theorem	nocvxminlem 27923*	Lemma for nocvxmin 27924. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴))) → ¬ 𝑋 <s 𝑌))
Theorem	nocvxmin 27924*	Given a nonempty convex class of surreals, there is a unique birthday-minimal element of that class. Lemma 0 of [Alling] p. 185. (Contributed by Scott Fenton, 30-Jun-2011.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∃!𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴))
Theorem	noprc 27925	The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ ¬ No ∈ V
15.3  Conway cut representation In [Conway] surreal numbers are represented as equivalence classes of cuts of previously defined surreal numbers. This is complicated to handle in ZFC without classes so we do not make it our definition. However, we can define a cut operator on surreals that behaves similarly. We introduce such an operator in this section and use it to define all surreals hearafter.
15.3.1  Conway cuts
Syntax	cslts 27926	Declare the syntax for surreal set less-than.
class <<s
Definition	df-slts 27927*	Define the relation that holds iff one set of surreals completely precedes another. (Contributed by Scott Fenton, 7-Dec-2021.)
⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
Syntax	ccuts 27928	Declare the syntax for the surreal cut operator.
class |s
Definition	df-cuts 27929*	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.)
⊢ |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)})))
Theorem	noeta2 27930*	A version of noeta 27883 with fewer hypotheses but a weaker upper bound (Contributed by Scott Fenton, 7-Dec-2021.)
⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
Theorem	brslts 27931*	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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
Theorem	sltsex1 27932	The first argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
Theorem	sltsex2 27933	The second argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
Theorem	sltsss1 27934	The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
Theorem	sltsss2 27935	The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
Theorem	sltssep 27936*	The separation property of surreal set less-than. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
Theorem	sltsd 27937*	Deduce surreal set less-than. (Contributed by Scott Fenton, 24-Sep-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ No )    &   ⊢ (𝜑 → 𝐵 ⊆ No )    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦)    ⇒   ⊢ (𝜑 → 𝐴 <<s 𝐵)
Theorem	sltssnb 27938	Surreal set less-than of two singletons. (Contributed by Scott Fenton, 18-Jan-2026.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))
Theorem	sltssn 27939	Surreal set less-than of two singletons. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    ⇒   ⊢ (𝜑 → {𝐴} <<s {𝐵})
Theorem	sltssepc 27940	Two elements of separated sets obey less-than. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌)
Theorem	sltssepcd 27941	Two elements of separated sets obey less-than. Deduction form of sltssepc 27940. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 <s 𝑌)
Theorem	ssslts1 27942	Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵)
Theorem	ssslts2 27943	Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶)
Theorem	nulslts 27944	The empty set is less-than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
Theorem	nulsgts 27945	The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)
Theorem	nulsltsd 27946	The empty set is less-than any set of surreals. Deduction version. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ No )    ⇒   ⊢ (𝜑 → ∅ <<s 𝐴)
Theorem	nulsgtsd 27947	The empty set is greater than any set of surreals. Deduction version. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ No )    ⇒   ⊢ (𝜑 → 𝐴 <<s ∅)
Theorem	conway 27948*	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. Theorem from [Alling] p. 185. (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)}))
Theorem	cutsval 27949*	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)})))
Theorem	cutcuts 27950	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))
Theorem	cutscl 27951	Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
Theorem	cutscld 27952	Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) ∈ No )
Theorem	cutbday 27953*	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)}))
Theorem	eqcuts 27954*	Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
Theorem	eqcuts2 27955*	Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑦)))))
Theorem	sltstr 27956	Transitive law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ∧ 𝐵 ≠ ∅) → 𝐴 <<s 𝐶)
Theorem	sltsun1 27957	Union law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)
Theorem	sltsun2 27958	Union law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))
Theorem	cutsun12 27959	Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (𝐴 |s 𝐵))
Theorem	dmcuts 27960	The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ dom |s = <<s
Theorem	cutsf 27961	Functionality statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
⊢ |s : <<s ⟶ No
Theorem	etaslts 27962*	A restatement of noeta 27883 using set less-than. (Contributed by Scott Fenton, 10-Aug-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))
Theorem	etaslts2 27963*	A version of etaslts 27962 with fewer hypotheses but a weaker upper bound. (Contributed by Scott Fenton, 10-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
Theorem	cutbdaybnd 27964	An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Aug-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)
Theorem	cutbdaybnd2 27965	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 “ (𝐴 ∪ 𝐵)))
Theorem	cutbdaybnd2lim 27966	An upper bound on the birthday of a surreal cut when it is a limit birthday. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ∪ ( bday “ (𝐴 ∪ 𝐵)))
Theorem	cutbdaylt 27967	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))
Theorem	lesrec 27968*	A comparison law for surreals considered as cuts of sets of surreals. Definition from [Conway] p. 4. Theorem 4 of [Alling] p. 186. Theorem 2.5 of [Gonshor] p. 9. (Contributed by Scott Fenton, 11-Dec-2021.)
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))
Theorem	lesrecd 27969*	A comparison law for surreals considered as cuts of sets of surreals. Definition from [Conway] p. 4. Theorem 4 of [Alling] p. 186. Theorem 2.5 of [Gonshor] p. 9. (Contributed by Scott Fenton, 5-Dec-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝐶 <<s 𝐷)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    &   ⊢ (𝜑 → 𝑌 = (𝐶 |s 𝐷))    ⇒   ⊢ (𝜑 → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))
Theorem	ltsrec 27970*	A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))
Theorem	ltsrecd 27971*	A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 5-Dec-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝐶 <<s 𝐷)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    &   ⊢ (𝜑 → 𝑌 = (𝐶 |s 𝐷))    ⇒   ⊢ (𝜑 → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))
Theorem	sltsdisj 27972	If 𝐴 preceeds 𝐵, then 𝐴 and 𝐵 are disjoint. (Contributed by Scott Fenton, 18-Sep-2024.)
⊢ (𝐴 <<s 𝐵 → (𝐴 ∩ 𝐵) = ∅)
Theorem	eqcuts3 27973*	A variant of the simplicity theorem - if 𝐵 lies between the cut sets of 𝐴 but none of its options do, then 𝐴 = 𝐵. Theorem 11 of [Conway] p. 23. (Contributed by Scott Fenton, 28-Nov-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝑀 <<s 𝑆)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))    &   ⊢ (𝜑 → 𝐿 <<s {𝐵})    &   ⊢ (𝜑 → {𝐵} <<s 𝑅)    &   ⊢ (𝜑 → ∀𝑥𝑂 ∈ (𝑀 ∪ 𝑆) ¬ (𝐿 <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s 𝑅))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
15.3.2  Zero and One
Syntax	c0s 27974	Declare the class syntax for surreal zero.
class 0s
Syntax	c1s 27975	Declare the class syntax for surreal one.
class 1s
Definition	df-0s 27976	Define surreal zero. This is the simplest cut of surreal number sets. Definition from [Conway] p. 17. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ 0s = (∅ |s ∅)
Definition	df-1s 27977	Define surreal one. This is the simplest number greater than surreal zero. Definition from [Conway] p. 18. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ 1s = ({ 0s } |s ∅)
Theorem	0no 27978	Surreal zero is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ 0s ∈ No
Theorem	1no 27979	Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ 1s ∈ No
Theorem	bday0 27980	Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ ( bday ‘ 0s ) = ∅
Theorem	0lt1s 27981	Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ 0s <s 1s
Theorem	bday0b 27982	The only surreal with birthday ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ ↔ 𝑋 = 0s ))
Theorem	bday1 27983	The birthday of surreal one is ordinal one. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ( bday ‘ 1s ) = 1o
Theorem	cuteq0 27984	Condition for a surreal cut to equal zero. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → 𝐴 <<s { 0s })    &   ⊢ (𝜑 → { 0s } <<s 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = 0s )
Theorem	cutneg 27985	The simplest number greater than a negative number is zero. (Contributed by Scott Fenton, 4-Sep-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 0s )    ⇒   ⊢ (𝜑 → ({𝐴} |s ∅) = 0s )
Theorem	cuteq1 27986	Condition for a surreal cut to equal one. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ (𝜑 → 0s ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 <<s { 1s })    &   ⊢ (𝜑 → { 1s } <<s 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = 1s )
Theorem	gt0ne0s 27987	A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ ( 0s <s 𝐴 → 𝐴 ≠ 0s )
Theorem	gt0ne0sd 27988	A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ (𝜑 → 0s <s 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0s )
Theorem	1ne0s 27989	Surreal zero does not equal surreal one. (Contributed by Scott Fenton, 5-Sep-2025.)
⊢ 1s ≠ 0s
Theorem	rightge0 27990*	A surreal is non-negative iff all its right options are positive. (Contributed by Scott Fenton, 1-Jan-2026.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   ⊢ (𝜑 → ( 0s ≤s 𝑋 ↔ ∀𝑥𝑅 ∈ 𝐵 0s <s 𝑥𝑅))
15.3.3  Cuts and Options
Syntax	cmade 27991	Declare the symbol for the made by function.
class M
Syntax	cold 27992	Declare the symbol for the older than function.
class O
Syntax	cnew 27993	Declare the symbol for the new on function.
class N
Syntax	cleft 27994	Declare the symbol for the left option function.
class L
Syntax	cright 27995	Declare the symbol for the right option function.
class R
Definition	df-made 27996	Define the made by function. This function carries an ordinal to all surreals made by sections of surreals older than it. Definition from [Conway] p. 29. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
Definition	df-old 27997	Define the older than function. This function carries an ordinal to all surreals made by a previous ordinal. Definition from [Conway] p. 29. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
Definition	df-new 27998	Define the newer than function. This function carries an ordinal to all surreals made on that day. Definition from [Conway] p. 29. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
Definition	df-left 27999*	Define the left options of a surreal. This is the set of surreals that are simpler and less than the given surreal. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
Definition	df-right 28000*	Define the right options of a surreal. This is the set of surreals that are simpler and greater than the given surreal. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦})