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	newf 27901	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 27902	No surreal is older than ∅. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ ( O ‘∅) = ∅
Theorem	madessno 27903	Made sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( M ‘𝐴) ⊆ No
Theorem	oldssno 27904	Old sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( O ‘𝐴) ⊆ No
Theorem	newssno 27905	New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( N ‘𝐴) ⊆ No
Theorem	madeno 27906	An element of a made set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( M ‘𝐵) → 𝐴 ∈ No )
Theorem	oldno 27907	An element of an old set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( O ‘𝐵) → 𝐴 ∈ No )
Theorem	newno 27908	An element of a new set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( N ‘𝐵) → 𝐴 ∈ No )
Theorem	madenod 27909	An element of a made set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( M ‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	oldnod 27910	An element of an old set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( O ‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	newnod 27911	An element of a new set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( N ‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	leftval 27912*	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 27913*	The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}
Theorem	elleft 27914	Membership in the left set of a surreal. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝐴 ∈ ( L ‘𝐵) ↔ (𝐴 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝐴 <s 𝐵))
Theorem	elright 27915	Membership in the right set of a surreal. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝐴 ∈ ( R ‘𝐵) ↔ (𝐴 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝐵 <s 𝐴))
Theorem	leftlt 27916	A member of a surreal's left set is less than it. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝐴 ∈ ( L ‘𝐵) → 𝐴 <s 𝐵)
Theorem	rightgt 27917	A member of a surreal's right set is greater than it. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝐴 ∈ ( R ‘𝐵) → 𝐵 <s 𝐴)
Theorem	leftf 27918	The functionality of the left options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ L : No ⟶𝒫 No
Theorem	rightf 27919	The functionality of the right options function. (Contributed by Scott Fenton, 6-Aug-2024.)
⊢ R : No ⟶𝒫 No
Theorem	elmade 27920*	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 27921*	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)))
Theorem	elold 27922*	Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
Theorem	sltsleft 27923	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})
Theorem	sltsright 27924	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))
Theorem	lltr 27925	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 ‘𝐴) <<s ( R ‘𝐴)
Theorem	made0 27926	The only surreal made on day ∅ is 0s. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ ( M ‘∅) = { 0s }
Theorem	new0 27927	The only surreal new on day ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ( N ‘∅) = { 0s }
Theorem	old1 27928	The only surreal older than 1o is 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( O ‘1o) = { 0s }
Theorem	madess 27929	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 ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
Theorem	oldssmade 27930	The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴)
Theorem	oldmade 27931	An element of an old set is an element of a made set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( O ‘𝐵) → 𝐴 ∈ ( M ‘𝐵))
Theorem	oldmaded 27932	An element of an old set is an element of a made set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( O ‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ∈ ( M ‘𝐵))
Theorem	oldss 27933	If 𝐴 is less than or equal to ordinal 𝐵, then the old set of 𝐴 is included in the made set of 𝐵. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( O ‘𝐴) ⊆ ( O ‘𝐵))
Theorem	leftssold 27934	The left options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( L ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
Theorem	rightssold 27935	The right options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( R ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
Theorem	leftssno 27936	The left set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( L ‘𝐴) ⊆ No
Theorem	rightssno 27937	The right set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ( R ‘𝐴) ⊆ No
Theorem	leftold 27938	An element of a left set is an element of the old set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( L ‘𝐵) → 𝐴 ∈ ( O ‘( bday ‘𝐵)))
Theorem	rightold 27939	An element of a right set is an element of the old set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( R ‘𝐵) → 𝐴 ∈ ( O ‘( bday ‘𝐵)))
Theorem	leftno 27940	An element of a left set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( L ‘𝐵) → 𝐴 ∈ No )
Theorem	rightno 27941	An element of a right set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( R ‘𝐵) → 𝐴 ∈ No )
Theorem	leftoldd 27942	An element of a left set is an element of the old set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( L ‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ∈ ( O ‘( bday ‘𝐵)))
Theorem	leftnod 27943	An element of a left set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( L ‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	rightoldd 27944	An element of a right set is an element of the old set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( R ‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ∈ ( O ‘( bday ‘𝐵)))
Theorem	rightnod 27945	An element of a right set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( R ‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	madecut 27946	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))
Theorem	madeun 27947	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))
Theorem	madeoldsuc 27948	The made set is the old set of its successor. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))
Theorem	oldsuc 27949	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)))
Theorem	oldlim 27950	The value of the old set at a limit ordinal. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → ( O ‘𝐴) = ∪ ( O “ 𝐴))
Theorem	madebdayim 27951	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)
Theorem	oldbdayim 27952	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)
Theorem	oldirr 27953	No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024.)
⊢ ¬ 𝑋 ∈ ( O ‘( bday ‘𝑋))
Theorem	leftirr 27954	No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ¬ 𝑋 ∈ ( L ‘𝑋)
Theorem	rightirr 27955	No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ ¬ 𝑋 ∈ ( R ‘𝑋)
Theorem	left0s 27956	The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( L ‘ 0s ) = ∅
Theorem	right0s 27957	The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ( R ‘ 0s ) = ∅
Theorem	left1s 27958	The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( L ‘ 1s ) = { 0s }
Theorem	right1s 27959	The right set of 1s is empty . (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( R ‘ 1s ) = ∅
Theorem	lrold 27960	The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.)
⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
Theorem	madebdaylemold 27961*	Lemma for madebday 27963. If the inductive hypothesis of madebday 27963 is satisfied, the converse of oldbdayim 27952 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))
Theorem	madebdaylemlrcut 27962*	Lemma for madebday 27963. If the inductive hypothesis of madebday 27963 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27967 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
Theorem	madebday 27963	A surreal is part of the set made by ordinal 𝐴 iff its birthday is less than or equal to 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))
Theorem	oldbday 27964	A surreal is part of the set older than ordinal 𝐴 iff its birthday is less than 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴))
Theorem	newbday 27965	A surreal is an element of a new set iff its birthday is equal to that ordinal. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴))
Theorem	newbdayim 27966	One direction of the biconditional in newbday 27965. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝑋 ∈ ( N ‘𝐴) → ( bday ‘𝑋) = 𝐴)
Theorem	lrcut 27967	A surreal is equal to the cut of its left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ (𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
Theorem	cutsfo 27968	The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ |s : <<s –onto→ No
Theorem	ltsn0 27969	If 𝑋 is less than 𝑌, then either ( L ‘𝑌) or ( R ‘𝑋) is non-empty. (Contributed by Scott Fenton, 10-Dec-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅))
Theorem	lruneq 27970	If two surreals share a birthday, then the union of their left and right sets are equal. (Contributed by Scott Fenton, 17-Sep-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
Theorem	ltslpss 27971	If two surreals share a birthday, then 𝑋 <s 𝑌 iff the left set of 𝑋 is a proper subset of the left set of 𝑌. (Contributed by Scott Fenton, 17-Sep-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))
Theorem	leslss 27972	If two surreals 𝐴 and 𝐵 share a birthday, then 𝐴 ≤s 𝐵 if and only if the left set of 𝐴 is a non-strict subset of the left set of 𝐵. (Contributed by Scott Fenton, 21-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 ≤s 𝐵 ↔ ( L ‘𝐴) ⊆ ( L ‘𝐵)))
Theorem	0elold 27973	Zero is in the old set of any nonzero number. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐴 ≠ 0s )    ⇒   ⊢ (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴)))
Theorem	0elleft 27974	Zero is in the left set of any positive number. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 0s <s 𝐴)    ⇒   ⊢ (𝜑 → 0s ∈ ( L ‘𝐴))
Theorem	0elright 27975	Zero is in the right set of any negative number. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 0s )    ⇒   ⊢ (𝜑 → 0s ∈ ( R ‘𝐴))
Theorem	madefi 27976	The made set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)
Theorem	oldfi 27977	The old set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin)
Theorem	bdayiun 27978*	The birthday of a surreal is the least upper bound of the successors of the birthdays of its options. This is the definition of the birthday of a combinatorial game in the Lean Combinatorial Game Theory library at https://github.com/vihdzp/combinatorial-games. (Contributed by Scott Fenton, 22-Nov-2025.)
⊢ (𝐴 ∈ No → ( bday ‘𝐴) = ∪ 𝑥 ∈ ( O ‘( bday ‘𝐴))suc ( bday ‘𝑥))
Theorem	bdayle 27979*	A condition for bounding a birthday above. (Contributed by Scott Fenton, 22-Nov-2025.)
⊢ ((𝑋 ∈ No ∧ Ord 𝑂) → (( bday ‘𝑋) ⊆ 𝑂 ↔ ∀𝑦 ∈ ( O ‘( bday ‘𝑋))( bday ‘𝑦) ∈ 𝑂))
Theorem	sltsbday 27980	Birthday comparison rule for surreals. (Contributed by Scott Fenton, 23-Feb-2026.)
⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐿 <<s {𝐵})    &   ⊢ (𝜑 → {𝐵} <<s 𝑅)    ⇒   ⊢ (𝜑 → ( bday ‘𝐴) ⊆ ( bday ‘𝐵))
15.3.4  Cofinality and coinitiality
Theorem	cofslts 27981*	If every element of 𝐴 is bounded above by some element of 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Note - we will often use the term "cofinal" to denote that every element of 𝐴 is bounded above by some element of 𝐵. Similarly, we will use the term "coinitial" to denote that every element of 𝐴 is bounded below by some element of 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)
Theorem	coinitslts 27982*	If 𝐵 is coinitial with 𝐶 and 𝐴 precedes 𝐶, then 𝐴 precedes 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
⊢ ((𝐵 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s 𝐵)
Theorem	cofcut1 27983*	If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcut1d 27984*	If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)    &   ⊢ (𝜑 → 𝐶 <<s {(𝐴 |s 𝐵)})    &   ⊢ (𝜑 → {(𝐴 |s 𝐵)} <<s 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcut2 27985*	If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcut2d 27986*	If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝒫 No )    &   ⊢ (𝜑 → 𝐷 ∈ 𝒫 No )    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)    &   ⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢)    &   ⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Theorem	cofcutr 27987*	If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋) and 𝐵 is coinitial with ( R ‘𝑋). Theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ ( R ‘𝑋)∃𝑤 ∈ 𝐵 𝑤 ≤s 𝑧))
Theorem	cofcutr1d 27988*	If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋). First half of theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦)
Theorem	cofcutr2d 27989*	If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐵 is coinitial with ( R ‘𝑋). Second half of theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   ⊢ (𝜑 → ∀𝑧 ∈ ( R ‘𝑋)∃𝑤 ∈ 𝐵 𝑤 ≤s 𝑧)
Theorem	cofcutrtime 27990*	If 𝑋 is the cut of 𝐴 and 𝐵 and all of 𝐴 and 𝐵 are older than 𝑋, then ( L ‘𝑋) is cofinal with 𝐴 and ( R ‘𝑋) is coinitial with 𝐵. Note: we will call a cut where all of the elements of the cut are older than the cut itself a "timely" cut. Part of Theorem 4.02(12) of [Alling] p. 125. (Contributed by Scott Fenton, 27-Sep-2024.)
⊢ (((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))
Theorem	cofcutrtime1d 27991*	If 𝑋 is a timely cut of 𝐴 and 𝐵, then ( L ‘𝑋) is cofinal with 𝐴. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)))    &   ⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
Theorem	cofcutrtime2d 27992*	If 𝑋 is a timely cut of 𝐴 and 𝐵, then ( R ‘𝑋) is coinitial with 𝐵. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)))    &   ⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))    ⇒   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
Theorem	cofss 27993*	Cofinality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ⊆ No )    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦)
Theorem	coiniss 27994*	Coinitiality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ⊆ No )    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)
Theorem	cutlt 27995*	Eliminating all elements below a given element of a cut does not affect the cut. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅)    &   ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐿)    ⇒   ⊢ (𝜑 → 𝐴 = (({𝑋} ∪ {𝑦 ∈ 𝐿 ∣ 𝑋 <s 𝑦}) |s 𝑅))
Theorem	cutpos 27996*	Reduce the elements of a cut for a positive number. (Contributed by Scott Fenton, 13-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 0s <s 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = (({ 0s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) |s ( R ‘𝐴)))
Theorem	cutmax 27997*	If 𝐴 has a maximum, then the maximum may be used alone in the cut. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑋)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = ({𝑋} |s 𝐵))
Theorem	cutmin 27998*	If 𝐵 has a minimum, then the minimum may be used alone in the cut. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝑋 ≤s 𝑦)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐴 |s {𝑋}))
Theorem	cutminmax 27999*	If the left set of 𝑋 has a maximum and the right set of 𝑋 has a minimum, then 𝑋 is equal to the cut of the maximum and the minimum. (Contributed by Scott Fenton, 23-Feb-2026.)
⊢ (𝜑 → 𝐿 ∈ ( L ‘𝑋))    &   ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)𝑥 ≤s 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ ( R ‘𝑋))    &   ⊢ (𝜑 → ∀𝑦 ∈ ( R ‘𝑋)𝑅 ≤s 𝑦)    ⇒   ⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅}))
15.4  Induction and recursion
15.4.1  Induction and recursion on one variable
Syntax	cnorec 28000	Declare the syntax for surreal recursion of one variable.
class norec (𝐹)