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Type | Label | Description |
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Statement | ||
Theorem | noetalem2 27801* | Lemma for noeta 27802. The full statement of the theorem with hypotheses in place. (Contributed by Scott Fenton, 10-Aug-2024.) |
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {〈dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑐 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂)) | ||
Theorem | noeta 27802* | 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, if the birthdays of members of 𝐴 and 𝐵 are strictly bounded above by 𝑂, then 𝑂 non-strictly bounds the separator. Axiom FE of [Alling] p. 185. (Contributed by Scott Fenton, 9-Aug-2024.) |
⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday ‘𝑧) ⊆ 𝑂)) | ||
Syntax | csle 27803 | Declare the syntax for surreal less-than or equal. |
class ≤s | ||
Definition | df-sle 27804 | Define the surreal less-than or equal predicate. Compare df-le 11298. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ≤s = (( No × No ) ∖ ◡ <s ) | ||
Theorem | sltirr 27805 | Surreal less-than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴) | ||
Theorem | slttr 27806 | Surreal less-than is transitive. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) | ||
Theorem | sltasym 27807 | Surreal less-than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴)) | ||
Theorem | sltlin 27808 | Surreal less-than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <s 𝐴)) | ||
Theorem | slttrieq2 27809 | 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))) | ||
Theorem | slttrine 27810 | Trichotomy law for surreals. (Contributed by Scott Fenton, 23-Nov-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≠ 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐵 <s 𝐴))) | ||
Theorem | slenlt 27811 | 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)) | ||
Theorem | sltnle 27812 | 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)) | ||
Theorem | sleloe 27813 | 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 𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | sletri3 27814 | Trichotomy law for surreal less-than or equal. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴))) | ||
Theorem | sltletr 27815 | 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 | slelttr 27816 | 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 | sletr 27817 | 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 | slttrd 27818 | 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 | sltletrd 27819 | 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 | slelttrd 27820 | 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 | sletrd 27821 | 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 27822 | 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 27823 | Surreal trichotomy law. (Contributed by Scott Fenton, 14-Feb-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ∨ 𝐵 ≤s 𝐴)) | ||
Theorem | maxs1 27824 | 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 27825 | 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 27826 | 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 27827 | 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 27828 | 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 27829 | Surreal less-than implies not equal. (Contributed by Scott Fenton, 12-Mar-2025.) |
⊢ ((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴) | ||
Theorem | sltlend 27830 | 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))) | ||
Theorem | bdayfun 27831 | The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.) |
⊢ Fun bday | ||
Theorem | bdayfn 27832 | The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.) |
⊢ bday Fn No | ||
Theorem | bdaydm 27833 | The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.) |
⊢ dom bday = No | ||
Theorem | bdayrn 27834 | The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.) |
⊢ ran bday = On | ||
Theorem | bdayelon 27835 | 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 27836* | Lemma for nocvxmin 27837. 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 27837* | 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 27838 | The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.) |
⊢ ¬ No ∈ V | ||
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. | ||
Syntax | csslt 27839 | Declare the syntax for surreal set less-than. |
class <<s | ||
Definition | df-sslt 27840* | 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 27841 | Declare the syntax for the surreal cut operator. |
class |s | ||
Definition | df-scut 27842* | 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 27843* | A version of noeta 27802 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 27844* | 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 27845 | The first argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | ||
Theorem | ssltex2 27846 | The second argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | ||
Theorem | ssltss1 27847 | 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 27848 | 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 27849* | 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 27850* | 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 27851 | 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 27852 | 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 27853 | Two elements of separated sets obey less-than. Deduction form of ssltsepc 27852. (Contributed by Scott Fenton, 25-Sep-2024.) |
⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 <s 𝑌) | ||
Theorem | sssslt1 27854 | 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 27855 | 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 27856 | The empty set is less-than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴) | ||
Theorem | nulssgt 27857 | The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅) | ||
Theorem | conway 27858* | 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 27859* | 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 27860 | 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 27861 | Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.) |
⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No ) | ||
Theorem | scutcld 27862 | Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.) |
⊢ (𝜑 → 𝐴 <<s 𝐵) ⇒ ⊢ (𝜑 → (𝐴 |s 𝐵) ∈ No ) | ||
Theorem | scutbday 27863* | 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 27864* | 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 27865* | 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 27866 | 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 27867 | 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 27868 | 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 27869 | 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 27870 | The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ dom |s = <<s | ||
Theorem | scutf 27871 | Functionality statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.) |
⊢ |s : <<s ⟶ No | ||
Theorem | etasslt 27872* | A restatement of noeta 27802 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 27873* | A version of etasslt 27872 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 27874 | 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 27875 | 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 27876 | 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 27877 | 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 27878* | 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 27879* | 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 27880 | If 𝐴 preceeds 𝐵, then 𝐴 and 𝐵 are disjoint. (Contributed by Scott Fenton, 18-Sep-2024.) |
⊢ (𝐴 <<s 𝐵 → (𝐴 ∩ 𝐵) = ∅) | ||
Syntax | c0s 27881 | Declare the class syntax for surreal zero. |
class 0s | ||
Syntax | c1s 27882 | Declare the class syntax for surreal one. |
class 1s | ||
Definition | df-0s 27883 | 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 27884 | 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 27885 | Surreal zero is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ 0s ∈ No | ||
Theorem | 1sno 27886 | Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ 1s ∈ No | ||
Theorem | bday0s 27887 | Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ ( bday ‘ 0s ) = ∅ | ||
Theorem | 0slt1s 27888 | Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.) |
⊢ 0s <s 1s | ||
Theorem | bday0b 27889 | The only surreal with birthday ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.) |
⊢ (𝑋 ∈ No → (( bday ‘𝑋) = ∅ ↔ 𝑋 = 0s )) | ||
Theorem | bday1s 27890 | The birthday of surreal one is ordinal one. (Contributed by Scott Fenton, 8-Aug-2024.) |
⊢ ( bday ‘ 1s ) = 1o | ||
Theorem | cuteq0 27891 | 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 27892 | 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 27893 | A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.) |
⊢ ( 0s <s 𝐴 → 𝐴 ≠ 0s ) | ||
Theorem | sgt0ne0d 27894 | A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.) |
⊢ (𝜑 → 0s <s 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0s ) | ||
Syntax | cmade 27895 | Declare the symbol for the made by function. |
class M | ||
Syntax | cold 27896 | Declare the symbol for the older than function. |
class O | ||
Syntax | cnew 27897 | Declare the symbol for the new on function. |
class N | ||
Syntax | cleft 27898 | Declare the symbol for the left option function. |
class L | ||
Syntax | cright 27899 | Declare the symbol for the right option function. |
class R | ||
Definition | df-made 27900 | 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)))) |
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