P. 276 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sletrd 27501	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	slerflex 27502	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	sletric 27503	Surreal trichotomy law. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ∨ 𝐵 ≤s 𝐴))
Theorem	maxs1 27504	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 27505	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 27506	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 27507	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	sltled 27508	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	sltne 27509	Surreal less-than implies not equal. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴)
15.2.2  Birthday Theorems
Theorem	bdayfun 27510	The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ Fun bday
Theorem	bdayfn 27511	The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.)
⊢ bday Fn No
Theorem	bdaydm 27512	The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ dom bday = No
Theorem	bdayrn 27513	The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ ran bday = On
Theorem	bdayelon 27514	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	nocvxminlem 27515*	Lemma for nocvxmin 27516. 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 27516*	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 27517	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	csslt 27518	Declare the syntax for surreal set less-than.
class <<s
Definition	df-sslt 27519*	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	cscut 27520	Declare the syntax for the surreal cut operator.
class |s
Definition	df-scut 27521*	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 27522*	A version of noeta 27482 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	brsslt 27523*	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	ssltex1 27524	The first argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
Theorem	ssltex2 27525	The second argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
Theorem	ssltss1 27526	The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
Theorem	ssltss2 27527	The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
Theorem	ssltsep 27528*	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	ssltd 27529*	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	ssltsn 27530	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	ssltsepc 27531	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	ssltsepcd 27532	Two elements of separated sets obey less-than. Deduction form of ssltsepc 27531. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 <s 𝑌)
Theorem	sssslt1 27533	Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵)
Theorem	sssslt2 27534	Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶)
Theorem	nulsslt 27535	The empty set is less-than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
Theorem	nulssgt 27536	The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)
Theorem	conway 27537*	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	scutval 27538*	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	scutcut 27539	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	scutcl 27540	Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
Theorem	scutcld 27541	Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) ∈ No )
Theorem	scutbday 27542*	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	eqscut 27543*	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	eqscut2 27544*	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	sslttr 27545	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	ssltun1 27546	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	ssltun2 27547	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	scutun12 27548	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	dmscut 27549	The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ dom |s = <<s
Theorem	scutf 27550	Functionality statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
⊢ |s : <<s ⟶ No
Theorem	etasslt 27551*	A restatement of noeta 27482 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	etasslt2 27552*	A version of etasslt 27551 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	scutbdaybnd 27553	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	scutbdaybnd2 27554	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	scutbdaybnd2lim 27555	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	scutbdaylt 27556	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	slerec 27557*	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	sltrec 27558*	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	ssltdisj 27559	If 𝐴 preceeds 𝐵, then 𝐴 and 𝐵 are disjoint. (Contributed by Scott Fenton, 18-Sep-2024.)
⊢ (𝐴 <<s 𝐵 → (𝐴 ∩ 𝐵) = ∅)
15.3.2  Zero and One
Syntax	c0s 27560	Declare the class syntax for surreal zero.
class 0s
Syntax	c1s 27561	Declare the class syntax for surreal one.
class 1s
Definition	df-0s 27562	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 27563	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	0sno 27564	Surreal zero is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ 0s ∈ No
Theorem	1sno 27565	Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ 1s ∈ No
Theorem	bday0s 27566	Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ ( bday ‘ 0s ) = ∅
Theorem	0slt1s 27567	Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ 0s <s 1s
Theorem	bday0b 27568	The only surreal with birthday ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ ↔ 𝑋 = 0s ))
Theorem	bday1s 27569	The birthday of surreal one is ordinal one. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ( bday ‘ 1s ) = 1o
Theorem	cuteq0 27570	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	cuteq1 27571	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	sgt0ne0 27572	A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ ( 0s <s 𝐴 → 𝐴 ≠ 0s )
Theorem	sgt0ne0d 27573	A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ (𝜑 → 0s <s 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0s )
15.3.3  Cuts and Options
Syntax	cmade 27574	Declare the symbol for the made by function.
class M
Syntax	cold 27575	Declare the symbol for the older than function.
class O
Syntax	cnew 27576	Declare the symbol for the new on function.
class N
Syntax	cleft 27577	Declare the symbol for the left option function.
class L
Syntax	cright 27578	Declare the symbol for the right option function.
class R
Definition	df-made 27579	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 27580	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 27581	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 27582*	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 27583*	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})
Theorem	madeval 27584	The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
Theorem	madeval2 27585*	Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
Theorem	oldval 27586	The value of the old options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
Theorem	newval 27587	The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
Theorem	madef 27588	The made function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ M :On⟶𝒫 No
Theorem	oldf 27589	The older function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ O :On⟶𝒫 No
Theorem	newf 27590	The new function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ N :On⟶𝒫 No
Theorem	old0 27591	No surreal is older than ∅. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ ( O ‘∅) = ∅
Theorem	madessno 27592	Made sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( M ‘𝐴) ⊆ No
Theorem	oldssno 27593	Old sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( O ‘𝐴) ⊆ No
Theorem	newssno 27594	New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( N ‘𝐴) ⊆ No
Theorem	leftval 27595*	The value of the left options function. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}
Theorem	rightval 27596*	The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}
Theorem	leftf 27597	The functionality of the left options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ L : No ⟶𝒫 No
Theorem	rightf 27598	The functionality of the right options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ R : No ⟶𝒫 No
Theorem	elmade 27599*	Membership in the made function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
Theorem	elmade2 27600*	Membership in the made function in terms of the old function. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))