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Theorem List for Metamath Proof Explorer - 33901-34000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnoprc 33901 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ No ∈ V
 
20.9.29  Surreal numbers: Conway cuts
 
Syntaxcsslt 33902 Declare the syntax for surreal set less than.
class <<s
 
Definitiondf-sslt 33903* 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 33904 Declare the syntax for the surreal cut operator.
class |s
 
Definitiondf-scut 33905* 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)})))
 
Theoremnoeta2 33906* A version of noeta 33873 with fewer hypotheses but a weaker upper bound (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))))
 
Theorembrsslt 33907* 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 33908 The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 ∈ V)
 
Theoremssltex2 33909 The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 ∈ V)
 
Theoremssltss1 33910 The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 No )
 
Theoremssltss2 33911 The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 No )
 
Theoremssltsep 33912* The separation property of surreal set less than. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
 
Theoremssltd 33913* Deduce surreal set less than. (Contributed by Scott Fenton, 24-Sep-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐴 No )    &   (𝜑𝐵 No )    &   ((𝜑𝑥𝐴𝑦𝐵) → 𝑥 <s 𝑦)       (𝜑𝐴 <<s 𝐵)
 
Theoremssltsepc 33914 Two elements of separated sets obey less than. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 <<s 𝐵𝑋𝐴𝑌𝐵) → 𝑋 <s 𝑌)
 
Theoremssltsepcd 33915 Two elements of separated sets obey less than. Deduction form of ssltsepc 33914. (Contributed by Scott Fenton, 25-Sep-2024.)
(𝜑𝐴 <<s 𝐵)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑𝑋 <s 𝑌)
 
Theoremsssslt1 33916 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)
 
Theoremsssslt2 33917 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)
 
Theoremnulsslt 33918 The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
 
Theoremnulssgt 33919 The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No 𝐴 <<s ∅)
 
Theoremconway 33920* 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)}))
 
Theoremscutval 33921* 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 33922 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))
 
Theoremscutcl 33923 Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
(𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
 
Theoremscutcld 33924 Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
(𝜑𝐴 <<s 𝐵)       (𝜑 → (𝐴 |s 𝐵) ∈ No )
 
Theoremscutbday 33925* 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)}))
 
Theoremeqscut 33926* Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
((𝐿 <<s 𝑅𝑋 No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday 𝑋) = ( bday “ {𝑦 No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
 
Theoremeqscut2 33927* Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
((𝐿 <<s 𝑅𝑋 No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday 𝑋) ⊆ ( bday 𝑦)))))
 
Theoremsslttr 33928 Transitive law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐵 <<s 𝐶𝐵 ≠ ∅) → 𝐴 <<s 𝐶)
 
Theoremssltun1 33929 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)
 
Theoremssltun2 33930 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
 
Theoremscutun12 33931 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 33932 The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
dom |s = <<s
 
Theoremscutf 33933 Functionality statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
|s : <<s ⟶ No
 
Theoremetasslt 33934* A restatement of noeta 33873 using set less than. (Contributed by Scott Fenton, 10-Aug-2024.)
((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ 𝑂))
 
Theoremetasslt2 33935* A version of etasslt 33934 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 “ (𝐴𝐵))))
 
Theoremscutbdaybnd 33936 An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Aug-2024.)
((𝐴 <<s 𝐵𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)
 
Theoremscutbdaybnd2 33937 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)))
 
Theoremscutbdaybnd2lim 33938 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 “ (𝐴𝐵)))
 
Theoremscutbdaylt 33939 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 33940* 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 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑𝐷 𝑋 <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s 𝑌)))
 
Theoremsltrec 33941* 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)))
 
Theoremssltdisj 33942 If 𝐴 preceeds 𝐵, then 𝐴 and 𝐵 are disjoint. (Contributed by Scott Fenton, 18-Sep-2024.)
(𝐴 <<s 𝐵 → (𝐴𝐵) = ∅)
 
20.9.30  Surreal numbers - zero and one
 
Syntaxc0s 33943 Declare the class syntax for surreal zero.
class 0s
 
Syntaxc1s 33944 Declare the class syntax for surreal one.
class 1s
 
Definitiondf-0s 33945 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 ∅)
 
Definitiondf-1s 33946 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 ∅)
 
Theorem0sno 33947 Surreal zero is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
0s ∈ No
 
Theorem1sno 33948 Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
1s ∈ No
 
Theorembday0s 33949 Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024.)
( bday ‘ 0s ) = ∅
 
Theorem0slt1s 33950 Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.)
0s <s 1s
 
Theorembday0b 33951 The only surreal with birthday is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝑋 No → (( bday 𝑋) = ∅ ↔ 𝑋 = 0s ))
 
Theorembday1s 33952 The birthday of surreal one is ordinal one. (Contributed by Scott Fenton, 8-Aug-2024.)
( bday ‘ 1s ) = 1o
 
20.9.31  Surreal numbers - cuts and options
 
Syntaxcmade 33953 Declare the symbol for the made by function.
class M
 
Syntaxcold 33954 Declare the symbol for the older than function.
class O
 
Syntaxcnew 33955 Declare the symbol for the new on function.
class N
 
Syntaxcleft 33956 Declare the symbol for the left option function.
class L
 
Syntaxcright 33957 Declare the symbol for the right option function.
class R
 
Definitiondf-made 33958 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))))
 
Definitiondf-old 33959 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))
 
Definitiondf-new 33960 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)))
 
Definitiondf-left 33961* 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})
 
Definitiondf-right 33962* 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})
 
Theoremmadeval 33963 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))))
 
Theoremmadeval2 33964* 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)})
 
Theoremoldval 33965 The value of the old options function. (Contributed by Scott Fenton, 6-Aug-2024.)
(𝐴 ∈ On → ( O ‘𝐴) = ( M “ 𝐴))
 
Theoremnewval 33966 The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
 
Theoremmadef 33967 The made function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
M :On⟶𝒫 No
 
Theoremoldf 33968 The older function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
O :On⟶𝒫 No
 
Theoremnewf 33969 The new function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
N :On⟶𝒫 No
 
Theoremold0 33970 No surreal is older than . (Contributed by Scott Fenton, 7-Aug-2024.)
( O ‘∅) = ∅
 
Theoremmadessno 33971 Made sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( M ‘𝐴) ⊆ No
 
Theoremoldssno 33972 Old sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( O ‘𝐴) ⊆ No
 
Theoremnewssno 33973 New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( N ‘𝐴) ⊆ No
 
Theoremleftval 33974* The value of the left options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴}
 
Theoremrightval 33975* The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥}
 
Theoremleftf 33976 The functionality of the left options function. (Contributed by Scott Fenton, 6-Aug-2024.)
L : No ⟶𝒫 No
 
Theoremrightf 33977 The functionality of the right options function. (Contributed by Scott Fenton, 6-Aug-2024.)
R : No ⟶𝒫 No
 
Theoremelmade 33978* 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)))
 
Theoremelmade2 33979* 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)))
 
Theoremelold 33980* Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
 
Theoremssltleft 33981 A surreal is greater than its left options. Theorem 0(ii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 No → ( L ‘𝐴) <<s {𝐴})
 
Theoremssltright 33982 A surreal is less than its right options. Theorem 0(i) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 No → {𝐴} <<s ( R ‘𝐴))
 
Theoremlltropt 33983 The left options of a surreal are strictly less than the right options of the same surreal. (Contributed by Scott Fenton, 6-Aug-2024.)
(𝐴 No → ( L ‘𝐴) <<s ( R ‘𝐴))
 
Theoremmade0 33984 The only surreal made on day is 0s. (Contributed by Scott Fenton, 7-Aug-2024.)
( M ‘∅) = { 0s }
 
Theoremnew0 33985 The only surreal new on day is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
( N ‘∅) = { 0s }
 
Theoremmadess 33986 If 𝐴 is less than or equal to ordinal 𝐵, then the made set of 𝐴 is included in the made set of 𝐵. (Contributed by Scott Fenton, 9-Oct-2024.)
((𝐵 ∈ On ∧ 𝐴𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
 
Theoremoldssmade 33987 The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.)
( O ‘𝐴) ⊆ ( M ‘𝐴)
 
Theoremleftssold 33988 The left options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
( L ‘𝑋) ⊆ ( O ‘( bday 𝑋))
 
Theoremrightssold 33989 The right options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
( R ‘𝑋) ⊆ ( O ‘( bday 𝑋))
 
Theoremleftssno 33990 The left set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( L ‘𝐴) ⊆ No
 
Theoremrightssno 33991 The right set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( R ‘𝐴) ⊆ No
 
Theoremmadecut 33992 Given a section that is a subset of an old set, the cut is a member of the made set. (Contributed by Scott Fenton, 7-Aug-2024.)
(((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) ∈ ( M ‘𝐴))
 
Theoremmadeun 33993 The made set is the union of the old set and the new set. (Contributed by Scott Fenton, 9-Oct-2024.)
( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴))
 
Theoremmadeoldsuc 33994 The made set is the old set of its successor. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))
 
Theoremoldsuc 33995 The value of the old set at a successor. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝐴 ∈ On → ( O ‘suc 𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴)))
 
Theoremoldlim 33996 The value of the old set at a limit ordinal. (Contributed by Scott Fenton, 8-Aug-2024.)
((Lim 𝐴𝐴𝑉) → ( O ‘𝐴) = ( O “ 𝐴))
 
Theoremmadebdayim 33997 If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)
(𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴)
 
Theoremoldbdayim 33998 If 𝑋 is in the old set for 𝐴, then the birthday of 𝑋 is less than 𝐴. (Contributed by Scott Fenton, 10-Aug-2024.)
(𝑋 ∈ ( O ‘𝐴) → ( bday 𝑋) ∈ 𝐴)
 
Theoremoldirr 33999 No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024.)
¬ 𝑋 ∈ ( O ‘( bday 𝑋))
 
Theoremleftirr 34000 No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.)
¬ 𝑋 ∈ ( L ‘𝑋)
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