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Theorem List for Metamath Proof Explorer - 27301-27400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssltsn 27301 Surreal set less-than of two singletons. (Contributed by Scott Fenton, 17-Mar-2025.)
(πœ‘ β†’ 𝐴 ∈ No )    &   (πœ‘ β†’ 𝐡 ∈ No )    &   (πœ‘ β†’ 𝐴 <s 𝐡)    β‡’   (πœ‘ β†’ {𝐴} <<s {𝐡})
 
Theoremssltsepc 27302 Two elements of separated sets obey less-than. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 <<s 𝐡 ∧ 𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 <s π‘Œ)
 
Theoremssltsepcd 27303 Two elements of separated sets obey less-than. Deduction form of ssltsepc 27302. (Contributed by Scott Fenton, 25-Sep-2024.)
(πœ‘ β†’ 𝐴 <<s 𝐡)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ 𝑋 <s π‘Œ)
 
Theoremsssslt1 27304 Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐡 ∧ 𝐢 βŠ† 𝐴) β†’ 𝐢 <<s 𝐡)
 
Theoremsssslt2 27305 Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐡 ∧ 𝐢 βŠ† 𝐡) β†’ 𝐴 <<s 𝐢)
 
Theoremnulsslt 27306 The empty set is less-than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No β†’ βˆ… <<s 𝐴)
 
Theoremnulssgt 27307 The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No β†’ 𝐴 <<s βˆ…)
 
Theoremconway 27308* Conway's Simplicity Theorem. Given 𝐴 preceeding 𝐡, 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 𝐡 β†’ βˆƒ!π‘₯ ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐡)} ( bday β€˜π‘₯) = ∩ ( bday β€œ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐡)}))
 
Theoremscutval 27309* The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐡 β†’ (𝐴 |s 𝐡) = (β„©π‘₯ ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐡)} ( bday β€˜π‘₯) = ∩ ( bday β€œ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐡)})))
 
Theoremscutcut 27310 Cut properties of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐡 β†’ ((𝐴 |s 𝐡) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐡)} ∧ {(𝐴 |s 𝐡)} <<s 𝐡))
 
Theoremscutcl 27311 Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
(𝐴 <<s 𝐡 β†’ (𝐴 |s 𝐡) ∈ No )
 
Theoremscutcld 27312 Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
(πœ‘ β†’ 𝐴 <<s 𝐡)    β‡’   (πœ‘ β†’ (𝐴 |s 𝐡) ∈ No )
 
Theoremscutbday 27313* The birthday of the surreal cut is equal to the minimum birthday in the gap. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐡 β†’ ( bday β€˜(𝐴 |s 𝐡)) = ∩ ( bday β€œ {π‘₯ ∈ No ∣ (𝐴 <<s {π‘₯} ∧ {π‘₯} <<s 𝐡)}))
 
Theoremeqscut 27314* Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) β†’ ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday β€˜π‘‹) = ∩ ( bday β€œ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
 
Theoremeqscut2 27315* Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) β†’ ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ βˆ€π‘¦ ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) β†’ ( bday β€˜π‘‹) βŠ† ( bday β€˜π‘¦)))))
 
Theoremsslttr 27316 Transitive law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐡 ∧ 𝐡 <<s 𝐢 ∧ 𝐡 β‰  βˆ…) β†’ 𝐴 <<s 𝐢)
 
Theoremssltun1 27317 Union law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐢 ∧ 𝐡 <<s 𝐢) β†’ (𝐴 βˆͺ 𝐡) <<s 𝐢)
 
Theoremssltun2 27318 Union law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐡 ∧ 𝐴 <<s 𝐢) β†’ 𝐴 <<s (𝐡 βˆͺ 𝐢))
 
Theoremscutun12 27319 Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐡 ∧ 𝐢 <<s {(𝐴 |s 𝐡)} ∧ {(𝐴 |s 𝐡)} <<s 𝐷) β†’ ((𝐴 βˆͺ 𝐢) |s (𝐡 βˆͺ 𝐷)) = (𝐴 |s 𝐡))
 
Theoremdmscut 27320 The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
dom |s = <<s
 
Theoremscutf 27321 Functionality statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
|s : <<s ⟢ No
 
Theoremetasslt 27322* A restatement of noeta 27253 using set less-than. (Contributed by Scott Fenton, 10-Aug-2024.)
((𝐴 <<s 𝐡 ∧ 𝑂 ∈ On ∧ ( bday β€œ (𝐴 βˆͺ 𝐡)) βŠ† 𝑂) β†’ βˆƒπ‘₯ ∈ No (𝐴 <<s {π‘₯} ∧ {π‘₯} <<s 𝐡 ∧ ( bday β€˜π‘₯) βŠ† 𝑂))
 
Theoremetasslt2 27323* A version of etasslt 27322 with fewer hypotheses but a weaker upper bound. (Contributed by Scott Fenton, 10-Dec-2021.)
(𝐴 <<s 𝐡 β†’ βˆƒπ‘₯ ∈ No (𝐴 <<s {π‘₯} ∧ {π‘₯} <<s 𝐡 ∧ ( bday β€˜π‘₯) βŠ† suc βˆͺ ( bday β€œ (𝐴 βˆͺ 𝐡))))
 
Theoremscutbdaybnd 27324 An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Aug-2024.)
((𝐴 <<s 𝐡 ∧ 𝑂 ∈ On ∧ ( bday β€œ (𝐴 βˆͺ 𝐡)) βŠ† 𝑂) β†’ ( bday β€˜(𝐴 |s 𝐡)) βŠ† 𝑂)
 
Theoremscutbdaybnd2 27325 An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.)
(𝐴 <<s 𝐡 β†’ ( bday β€˜(𝐴 |s 𝐡)) βŠ† suc βˆͺ ( bday β€œ (𝐴 βˆͺ 𝐡)))
 
Theoremscutbdaybnd2lim 27326 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 𝐡 ∧ Lim ( bday β€˜(𝐴 |s 𝐡))) β†’ ( bday β€˜(𝐴 |s 𝐡)) βŠ† βˆͺ ( bday β€œ (𝐴 βˆͺ 𝐡)))
 
Theoremscutbdaylt 27327 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 𝐡) ∧ 𝑋 β‰  (𝐴 |s 𝐡)) β†’ ( bday β€˜(𝐴 |s 𝐡)) ∈ ( bday β€˜π‘‹))
 
Theoremslerec 27328* 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 𝐡 ∧ 𝐢 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐡) ∧ π‘Œ = (𝐢 |s 𝐷))) β†’ (𝑋 ≀s π‘Œ ↔ (βˆ€π‘‘ ∈ 𝐷 𝑋 <s 𝑑 ∧ βˆ€π‘Ž ∈ 𝐴 π‘Ž <s π‘Œ)))
 
Theoremsltrec 27329* A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)
(((𝐴 <<s 𝐡 ∧ 𝐢 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐡) ∧ π‘Œ = (𝐢 |s 𝐷))) β†’ (𝑋 <s π‘Œ ↔ (βˆƒπ‘ ∈ 𝐢 𝑋 ≀s 𝑐 ∨ βˆƒπ‘ ∈ 𝐡 𝑏 ≀s π‘Œ)))
 
Theoremssltdisj 27330 If 𝐴 preceeds 𝐡, then 𝐴 and 𝐡 are disjoint. (Contributed by Scott Fenton, 18-Sep-2024.)
(𝐴 <<s 𝐡 β†’ (𝐴 ∩ 𝐡) = βˆ…)
 
15.3.2  Zero and One
 
Syntaxc0s 27331 Declare the class syntax for surreal zero.
class 0s
 
Syntaxc1s 27332 Declare the class syntax for surreal one.
class 1s
 
Definitiondf-0s 27333 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 = (βˆ… |s βˆ…)
 
Definitiondf-1s 27334 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 βˆ…)
 
Theorem0sno 27335 Surreal zero is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
0s ∈ No
 
Theorem1sno 27336 Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
1s ∈ No
 
Theorembday0s 27337 Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024.)
( bday β€˜ 0s ) = βˆ…
 
Theorem0slt1s 27338 Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.)
0s <s 1s
 
Theorembday0b 27339 The only surreal with birthday βˆ… is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝑋 ∈ No β†’ (( bday β€˜π‘‹) = βˆ… ↔ 𝑋 = 0s ))
 
Theorembday1s 27340 The birthday of surreal one is ordinal one. (Contributed by Scott Fenton, 8-Aug-2024.)
( bday β€˜ 1s ) = 1o
 
Theoremcuteq0 27341 Condition for a surreal cut to equal zero. (Contributed by Scott Fenton, 3-Feb-2025.)
(πœ‘ β†’ 𝐴 <<s { 0s })    &   (πœ‘ β†’ { 0s } <<s 𝐡)    β‡’   (πœ‘ β†’ (𝐴 |s 𝐡) = 0s )
 
Theoremcuteq1 27342 Condition for a surreal cut to equal one. (Contributed by Scott Fenton, 12-Mar-2025.)
(πœ‘ β†’ 0s ∈ 𝐴)    &   (πœ‘ β†’ 𝐴 <<s { 1s })    &   (πœ‘ β†’ { 1s } <<s 𝐡)    β‡’   (πœ‘ β†’ (𝐴 |s 𝐡) = 1s )
 
Theoremsgt0ne0 27343 A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.)
( 0s <s 𝐴 β†’ 𝐴 β‰  0s )
 
Theoremsgt0ne0d 27344 A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.)
(πœ‘ β†’ 0s <s 𝐴)    β‡’   (πœ‘ β†’ 𝐴 β‰  0s )
 
15.3.3  Cuts and Options
 
Syntaxcmade 27345 Declare the symbol for the made by function.
class M
 
Syntaxcold 27346 Declare the symbol for the older than function.
class O
 
Syntaxcnew 27347 Declare the symbol for the new on function.
class N
 
Syntaxcleft 27348 Declare the symbol for the left option function.
class L
 
Syntaxcright 27349 Declare the symbol for the right option function.
class R
 
Definitiondf-made 27350 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 β€œ (𝒫 βˆͺ ran 𝑓 Γ— 𝒫 βˆͺ ran 𝑓))))
 
Definitiondf-old 27351 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 = (π‘₯ ∈ On ↦ βˆͺ ( M β€œ π‘₯))
 
Definitiondf-new 27352 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 = (π‘₯ ∈ On ↦ (( M β€˜π‘₯) βˆ– ( O β€˜π‘₯)))
 
Definitiondf-left 27353* 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 = (π‘₯ ∈ No ↦ {𝑦 ∈ ( O β€˜( bday β€˜π‘₯)) ∣ 𝑦 <s π‘₯})
 
Definitiondf-right 27354* 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 = (π‘₯ ∈ No ↦ {𝑦 ∈ ( O β€˜( bday β€˜π‘₯)) ∣ π‘₯ <s 𝑦})
 
Theoremmadeval 27355 The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
(𝐴 ∈ On β†’ ( M β€˜π΄) = ( |s β€œ (𝒫 βˆͺ ( M β€œ 𝐴) Γ— 𝒫 βˆͺ ( M β€œ 𝐴))))
 
Theoremmadeval2 27356* Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
(𝐴 ∈ On β†’ ( M β€˜π΄) = {π‘₯ ∈ No ∣ βˆƒπ‘Ž ∈ 𝒫 βˆͺ ( M β€œ 𝐴)βˆƒπ‘ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(π‘Ž <<s 𝑏 ∧ (π‘Ž |s 𝑏) = π‘₯)})
 
Theoremoldval 27357 The value of the old options function. (Contributed by Scott Fenton, 6-Aug-2024.)
(𝐴 ∈ On β†’ ( O β€˜π΄) = βˆͺ ( M β€œ 𝐴))
 
Theoremnewval 27358 The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( N β€˜π΄) = (( M β€˜π΄) βˆ– ( O β€˜π΄))
 
Theoremmadef 27359 The made function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
M :OnβŸΆπ’« No
 
Theoremoldf 27360 The older function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
O :OnβŸΆπ’« No
 
Theoremnewf 27361 The new function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
N :OnβŸΆπ’« No
 
Theoremold0 27362 No surreal is older than βˆ…. (Contributed by Scott Fenton, 7-Aug-2024.)
( O β€˜βˆ…) = βˆ…
 
Theoremmadessno 27363 Made sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( M β€˜π΄) βŠ† No
 
Theoremoldssno 27364 Old sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( O β€˜π΄) βŠ† No
 
Theoremnewssno 27365 New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( N β€˜π΄) βŠ† No
 
Theoremleftval 27366* The value of the left options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( L β€˜π΄) = {π‘₯ ∈ ( O β€˜( bday β€˜π΄)) ∣ π‘₯ <s 𝐴}
 
Theoremrightval 27367* The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( R β€˜π΄) = {π‘₯ ∈ ( O β€˜( bday β€˜π΄)) ∣ 𝐴 <s π‘₯}
 
Theoremleftf 27368 The functionality of the left options function. (Contributed by Scott Fenton, 6-Aug-2024.)
L : No βŸΆπ’« No
 
Theoremrightf 27369 The functionality of the right options function. (Contributed by Scott Fenton, 6-Aug-2024.)
R : No βŸΆπ’« No
 
Theoremelmade 27370* Membership in the made function. (Contributed by Scott Fenton, 6-Aug-2024.)
(𝐴 ∈ On β†’ (𝑋 ∈ ( M β€˜π΄) ↔ βˆƒπ‘™ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)βˆƒπ‘Ÿ ∈ 𝒫 βˆͺ ( M β€œ 𝐴)(𝑙 <<s π‘Ÿ ∧ (𝑙 |s π‘Ÿ) = 𝑋)))
 
Theoremelmade2 27371* Membership in the made function in terms of the old function. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ On β†’ (𝑋 ∈ ( M β€˜π΄) ↔ βˆƒπ‘™ ∈ 𝒫 ( O β€˜π΄)βˆƒπ‘Ÿ ∈ 𝒫 ( O β€˜π΄)(𝑙 <<s π‘Ÿ ∧ (𝑙 |s π‘Ÿ) = 𝑋)))
 
Theoremelold 27372* Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ On β†’ (𝑋 ∈ ( O β€˜π΄) ↔ βˆƒπ‘ ∈ 𝐴 𝑋 ∈ ( M β€˜π‘)))
 
Theoremssltleft 27373 A surreal is greater than its left options. Theorem 0(ii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ No β†’ ( L β€˜π΄) <<s {𝐴})
 
Theoremssltright 27374 A surreal is less than its right options. Theorem 0(i) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ No β†’ {𝐴} <<s ( R β€˜π΄))
 
Theoremlltropt 27375 The left options of a surreal are strictly less than the right options of the same surreal. (Contributed by Scott Fenton, 6-Aug-2024.) (Revised by Scott Fenton, 21-Feb-2025.)
( L β€˜π΄) <<s ( R β€˜π΄)
 
Theoremmade0 27376 The only surreal made on day βˆ… is 0s. (Contributed by Scott Fenton, 7-Aug-2024.)
( M β€˜βˆ…) = { 0s }
 
Theoremnew0 27377 The only surreal new on day βˆ… is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
( N β€˜βˆ…) = { 0s }
 
Theoremold1 27378 The only surreal older than 1o is 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
( O β€˜1o) = { 0s }
 
Theoremmadess 27379 If 𝐴 is less than or equal to ordinal 𝐡, then the made set of 𝐴 is included in the made set of 𝐡. (Contributed by Scott Fenton, 9-Oct-2024.)
((𝐡 ∈ On ∧ 𝐴 βŠ† 𝐡) β†’ ( M β€˜π΄) βŠ† ( M β€˜π΅))
 
Theoremoldssmade 27380 The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.)
( O β€˜π΄) βŠ† ( M β€˜π΄)
 
Theoremleftssold 27381 The left options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
( L β€˜π‘‹) βŠ† ( O β€˜( bday β€˜π‘‹))
 
Theoremrightssold 27382 The right options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
( R β€˜π‘‹) βŠ† ( O β€˜( bday β€˜π‘‹))
 
Theoremleftssno 27383 The left set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( L β€˜π΄) βŠ† No
 
Theoremrightssno 27384 The right set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( R β€˜π΄) βŠ† No
 
Theoremmadecut 27385 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 βŠ† ( O β€˜π΄) ∧ 𝑅 βŠ† ( O β€˜π΄))) β†’ (𝐿 |s 𝑅) ∈ ( M β€˜π΄))
 
Theoremmadeun 27386 The made set is the union of the old set and the new set. (Contributed by Scott Fenton, 9-Oct-2024.)
( M β€˜π΄) = (( O β€˜π΄) βˆͺ ( N β€˜π΄))
 
Theoremmadeoldsuc 27387 The made set is the old set of its successor. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝐴 ∈ On β†’ ( M β€˜π΄) = ( O β€˜suc 𝐴))
 
Theoremoldsuc 27388 The value of the old set at a successor. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝐴 ∈ On β†’ ( O β€˜suc 𝐴) = (( O β€˜π΄) βˆͺ ( N β€˜π΄)))
 
Theoremoldlim 27389 The value of the old set at a limit ordinal. (Contributed by Scott Fenton, 8-Aug-2024.)
((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) β†’ ( O β€˜π΄) = βˆͺ ( O β€œ 𝐴))
 
Theoremmadebdayim 27390 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 β€˜π΄) β†’ ( bday β€˜π‘‹) βŠ† 𝐴)
 
Theoremoldbdayim 27391 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 β€˜π΄) β†’ ( bday β€˜π‘‹) ∈ 𝐴)
 
Theoremoldirr 27392 No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024.)
Β¬ 𝑋 ∈ ( O β€˜( bday β€˜π‘‹))
 
Theoremleftirr 27393 No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.)
Β¬ 𝑋 ∈ ( L β€˜π‘‹)
 
Theoremrightirr 27394 No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.)
Β¬ 𝑋 ∈ ( R β€˜π‘‹)
 
Theoremleft0s 27395 The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
( L β€˜ 0s ) = βˆ…
 
Theoremright0s 27396 The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
( R β€˜ 0s ) = βˆ…
 
Theoremleft1s 27397 The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
( L β€˜ 1s ) = { 0s }
 
Theoremright1s 27398 The right set of 1s is empty . (Contributed by Scott Fenton, 4-Feb-2025.)
( R β€˜ 1s ) = βˆ…
 
Theoremlrold 27399 The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.)
(( L β€˜π΄) βˆͺ ( R β€˜π΄)) = ( O β€˜( bday β€˜π΄))
 
Theoremmadebdaylemold 27400* Lemma for madebday 27402. If the inductive hypothesis of madebday 27402 is satisfied, the converse of oldbdayim 27391 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
((𝐴 ∈ On ∧ βˆ€π‘ ∈ 𝐴 βˆ€π‘¦ ∈ No (( bday β€˜π‘¦) βŠ† 𝑏 β†’ 𝑦 ∈ ( M β€˜π‘)) ∧ 𝑋 ∈ No ) β†’ (( bday β€˜π‘‹) ∈ 𝐴 β†’ 𝑋 ∈ ( O β€˜π΄)))
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