P. 285 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	n0sge0 28401	A non-negative integer is greater than or equal to zero. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
Theorem	nnsgt0 28402	A positive integer is greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 0s <s 𝐴)
Theorem	elnns 28403	Membership in the positive surreal integers. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
Theorem	elnns2 28404	A positive surreal integer is a non-negative surreal integer greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
Theorem	n0s0suc 28405*	A non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-Jul-2025.)
⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))
Theorem	nnsge1 28406	A positive surreal integer is greater than or equal to one. (Contributed by Scott Fenton, 26-Jul-2025.)
⊢ (𝑁 ∈ ℕs → 1s ≤s 𝑁)
Theorem	n0addscl 28407	The non-negative surreal integers are closed under addition. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 +s 𝐵) ∈ ℕ0s)
Theorem	n0mulscl 28408	The non-negative surreal integers are closed under multiplication. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕ0s) → (𝐴 ·s 𝐵) ∈ ℕ0s)
Theorem	nnaddscl 28409	The positive surreal integers are closed under addition. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) → (𝐴 +s 𝐵) ∈ ℕs)
Theorem	nnmulscl 28410	The positive surreal integers are closed under multiplication. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) → (𝐴 ·s 𝐵) ∈ ℕs)
Theorem	1n0s 28411	Surreal one is a non-negative surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ 1s ∈ ℕ0s
Theorem	1nns 28412	Surreal one is a positive surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ 1s ∈ ℕs
Theorem	peano2nns 28413	Peano postulate for positive surreal integers. One plus a positive surreal integer is a positive surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → (𝐴 +s 1s ) ∈ ℕs)
Theorem	nnsrecgt0d 28414	The reciprocal of a positive surreal integer is positive. (Contributed by Scott Fenton, 19-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕs)    ⇒   ⊢ (𝜑 → 0s <s ( 1s /su 𝐴))
Theorem	n0bday 28415	A non-negative surreal integer has a finite birthday. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)
Theorem	n0ssoldg 28416	The non-negative surreal integers are a subset of the old set of ω. To avoid the axiom of infinity, we include it as an antecedent. (Contributed by Scott Fenton, 20-Feb-2026.)
⊢ (ω ∈ V → ℕ0s ⊆ ( O ‘ω))
Theorem	n0ssold 28417	The non-negative surreal integers are a subset of the old set of ω. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ ℕ0s ⊆ ( O ‘ω)
Theorem	n0fincut 28418	The simplest number greater than a finite set of non-negative surreal integers is a non-negative surreal integer. (Contributed by Scott Fenton, 5-Nov-2025.)
⊢ ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s)
Theorem	onsfi 28419	A surreal ordinal with a finite birthday is a non-negative surreal integer. (Contributed by Scott Fenton, 4-Nov-2025.)
⊢ ((𝐴 ∈ Ons ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℕ0s)
Theorem	eln0s2 28420	A non-negative surreal integer is a surreal ordinal with a finite birthday. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ Ons ∧ ( bday ‘𝐴) ∈ ω))
Theorem	onltn0s 28421	A surreal ordinal that is less than a non-negative integer is a non-negative integer. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ℕ0s ∧ 𝐴 <s 𝐵) → 𝐴 ∈ ℕ0s)
Theorem	n0cutlt 28422*	A non-negative surreal integer is the simplest number greater than all previous non-negative surreal integers. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({𝑥 ∈ ℕ0s ∣ 𝑥 <s 𝐴} |s ∅))
Theorem	seqn0sfn 28423	The surreal sequence builder is a function over ℕ0s when started from zero. (Contributed by Scott Fenton, 19-Apr-2025.)
⊢ (𝜑 → seqs 0s ( + , 𝐹) Fn ℕ0s)
Theorem	eln0s 28424	A non-negative surreal integer is zero or a positive surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕs ∨ 𝐴 = 0s ))
Theorem	n0s0m1 28425	Every non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s))
Theorem	n0subs 28426	Subtraction of non-negative surreal integers. (Contributed by Scott Fenton, 26-May-2025.)
⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))
Theorem	n0subs2 28427	Subtraction of non-negative surreal integers. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕs))
Theorem	n0ltsp1le 28428	Non-negative surreal ordering relation. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑀 +s 1s ) ≤s 𝑁))
Theorem	n0lesltp1 28429	Non-negative surreal ordering relation. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ 𝑀 <s (𝑁 +s 1s )))
Theorem	n0lesm1lt 28430	Non-negative surreal ordering relation. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑀 -s 1s ) <s 𝑁))
Theorem	n0lts1e0 28431	A non-negative surreal integer is less than one iff it is zero. (Contributed by Scott Fenton, 23-Feb-2026.)
⊢ (𝐴 ∈ ℕ0s → (𝐴 <s 1s ↔ 𝐴 = 0s ))
Theorem	bdayn0p1 28432	The birthday of 𝐴 +s 1s is the successor of the birthday of 𝐴 when 𝐴 is a non-negative surreal integer. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = suc ( bday ‘𝐴))
Theorem	bdayn0sf1o 28433	The birthday function restricted to the non-negative surreal integers is a bijection with the finite ordinals. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
Theorem	n0p1nns 28434	One plus a non-negative surreal integer is a positive surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
Theorem	dfnns2 28435	Alternate definition of the positive surreal integers. Compare df-nn 12201. (Contributed by Scott Fenton, 6-Aug-2025.)
⊢ ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
Theorem	nnsind 28436*	Principle of Mathematical Induction (inference schema). (Contributed by Scott Fenton, 6-Aug-2025.)
⊢ (𝑥 = 1s → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 +s 1s ) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕs → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ℕs → 𝜏)
Theorem	nn1m1nns 28437	Every positive surreal integer is either one or a successor. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ (𝐴 ∈ ℕs → (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs))
Theorem	nnm1n0s 28438	A positive surreal integer minus one is a non-negative surreal integer. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ (𝑁 ∈ ℕs → (𝑁 -s 1s ) ∈ ℕ0s)
Theorem	eucliddivs 28439*	Euclid's division lemma for surreal numbers. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))
Theorem	oldfib 28440	The old set of an ordinal is finite iff the ordinal is finite. (Contributed by Scott Fenton, 19-Feb-2026.)
⊢ (𝐴 ∈ On → (𝐴 ∈ ω ↔ ( O ‘𝐴) ∈ Fin))
15.6.4  Integers
Syntax	czs 28441	Declare the syntax for surreal integers.
class ℤs
Definition	df-zs 28442	Define the surreal integers. Compare dfz2 12577. (Contributed by Scott Fenton, 17-May-2025.)
⊢ ℤs = ( -s “ (ℕs × ℕs))
Theorem	zsex 28443	The surreal integers form a set. (Contributed by Scott Fenton, 17-May-2025.)
⊢ ℤs ∈ V
Theorem	zssno 28444	The surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-May-2025.)
⊢ ℤs ⊆ No
Theorem	zno 28445	A surreal integer is a surreal. (Contributed by Scott Fenton, 17-May-2025.)
⊢ (𝐴 ∈ ℤs → 𝐴 ∈ No )
Theorem	znod 28446	A surreal integer is a surreal. Deduction form. (Contributed by Scott Fenton, 17-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℤs)    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	elzs 28447*	Membership in the set of surreal integers. (Contributed by Scott Fenton, 17-May-2025.)
⊢ (𝐴 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦))
Theorem	nnzsubs 28448	The difference of two surreal positive integers is an integer. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ ((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) → (𝐴 -s 𝐵) ∈ ℤs)
Theorem	nnzs 28449	A positive surreal integer is a surreal integer. (Contributed by Scott Fenton, 17-May-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℤs)
Theorem	nnzsd 28450	A positive surreal integer is a surreal integer. Deduction form. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕs)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤs)
Theorem	0zs 28451	Zero is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ 0s ∈ ℤs
Theorem	n0zs 28452	A non-negative surreal integer is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℤs)
Theorem	n0zsd 28453	A non-negative surreal integer is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤs)
Theorem	1zs 28454	One is a surreal integer. (Contributed by Scott Fenton, 24-Jul-2025.)
⊢ 1s ∈ ℤs
Theorem	znegscl 28455	The surreal integers are closed under negation. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℤs → ( -us ‘𝐴) ∈ ℤs)
Theorem	znegscld 28456	The surreal integers are closed under negation. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℤs)    ⇒   ⊢ (𝜑 → ( -us ‘𝐴) ∈ ℤs)
Theorem	zaddscl 28457	The surreal integers are closed under addition. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ ((𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs) → (𝐴 +s 𝐵) ∈ ℤs)
Theorem	zaddscld 28458	The surreal integers are closed under addition. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℤs)    &   ⊢ (𝜑 → 𝐵 ∈ ℤs)    ⇒   ⊢ (𝜑 → (𝐴 +s 𝐵) ∈ ℤs)
Theorem	zsubscld 28459	The surreal integers are closed under subtraction. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℤs)    &   ⊢ (𝜑 → 𝐵 ∈ ℤs)    ⇒   ⊢ (𝜑 → (𝐴 -s 𝐵) ∈ ℤs)
Theorem	zmulscld 28460	The surreal integers are closed under multiplication. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ ℤs)    &   ⊢ (𝜑 → 𝐵 ∈ ℤs)    ⇒   ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ ℤs)
Theorem	elzn0s 28461	A surreal integer is a surreal that is a non-negative integer or whose negative is a non-negative integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
Theorem	elzs2 28462	A surreal integer is either a positive integer, zero, or the negative of a positive integer. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs)))
Theorem	eln0zs 28463	Non-negative surreal integer property expressed in terms of integers. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
Theorem	elnnzs 28464	Positive surreal integer property expressed in terms of integers. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℤs ∧ 0s <s 𝑁))
Theorem	elznns 28465	Surreal integer property expressed in terms of positive integers and non-negative integers. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕ0s)))
Theorem	zn0subs 28466	The non-negative difference of surreal integers is a non-negative integer. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))
Theorem	peano5uzs 28467*	Peano's inductive postulate for upper surreal integers. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ (𝜑 → 𝑁 ∈ ℤs)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 +s 1s ) ∈ 𝐴)    ⇒   ⊢ (𝜑 → {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ⊆ 𝐴)
Theorem	uzsind 28468*	Induction on the upper surreal integers that start at 𝑀. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 +s 1s ) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑀 ∈ ℤs → 𝜓)    &   ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → 𝜏)
Theorem	zsbday 28469	A surreal integer has a finite birthday. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℤs → ( bday ‘𝐴) ∈ ω)
Theorem	zcuts 28470	A cut expression for surreal integers. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ (𝐴 ∈ ℤs → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
Theorem	zcuts0 28471	Either the left or right set of a surreal integer is empty. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝐴 ∈ ℤs → (( L ‘𝐴) = ∅ ∨ ( R ‘𝐴) = ∅))
Theorem	zsoring 28472	The surreal integers form an ordered ring. Note that we have to restrict the operations here since No is a proper class. (Contributed by Scott Fenton, 23-Dec-2025.)
⊢ ℤs = (Base‘𝐾)    &   ⊢ ( +s ↾ (ℤs × ℤs)) = (+g‘𝐾)    &   ⊢ ( ·s ↾ (ℤs × ℤs)) = (.r‘𝐾)    &   ⊢ ( ≤s ∩ (ℤs × ℤs)) = (le‘𝐾)    &   ⊢ 0s = (0g‘𝐾)    ⇒   ⊢ 𝐾 ∈ oRing
15.6.5  Dyadic fractions
Syntax	c2s 28473	Declare the syntax for surreal two.
class 2s
Definition	df-2s 28474	Define surreal two. This is the simplest number greater than one. See 1p1e2s 28479 for its addition version. (Contributed by Scott Fenton, 27-May-2025.)
⊢ 2s = ({ 1s } |s ∅)
Syntax	cexps 28475	Declare the syntax for surreal exponentiation.
class ↑s
Definition	df-exps 28476*	Define surreal exponentiation. Compare df-exp 14065. (Contributed by Scott Fenton, 27-May-2025.)
⊢ ↑s = (𝑥 ∈ No , 𝑦 ∈ ℤs ↦ if(𝑦 = 0s , 1s , if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦))))))
Syntax	cz12s 28477	Define the syntax for the set of surreal dyadic fractions.
class ℤs[1/2]
Definition	df-z12s 28478*	Define the set of dyadic rationals. This is the set of rationals whose denominator is a power of two. Later we will prove that this is precisely the set of surreals with a finite birthday. (Contributed by Scott Fenton, 27-May-2025.)
⊢ ℤs[1/2] = {𝑥 ∣ ∃𝑦 ∈ ℤs ∃𝑧 ∈ ℕ0s 𝑥 = (𝑦 /su (2s↑s𝑧))}
Theorem	1p1e2s 28479	One plus one is two. Surreal version. (Contributed by Scott Fenton, 27-May-2025.)
⊢ ( 1s +s 1s ) = 2s
Theorem	no2times 28480	Version of 2times 12343 for surreal numbers. (Contributed by Scott Fenton, 23-Jul-2025.)
⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
Theorem	2nns 28481	Surreal two is a surreal natural. (Contributed by Scott Fenton, 23-Jul-2025.)
⊢ 2s ∈ ℕs
Theorem	2no 28482	Surreal two is a surreal number. (Contributed by Scott Fenton, 23-Jul-2025.)
⊢ 2s ∈ No
Theorem	2ne0s 28483	Surreal two is nonzero. (Contributed by Scott Fenton, 23-Jul-2025.)
⊢ 2s ≠ 0s
Theorem	n0seo 28484*	A non-negative surreal integer is either even or odd. (Contributed by Scott Fenton, 19-Aug-2025.)
⊢ (𝑁 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))
Theorem	zseo 28485*	A surreal integer is either even or odd. (Contributed by Scott Fenton, 19-Aug-2025.)
⊢ (𝑁 ∈ ℤs → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))
Theorem	twocut 28486	Two times the cut of zero and one is one. (Contributed by Scott Fenton, 5-Sep-2025.)
⊢ (2s ·s ({ 0s } |s { 1s })) = 1s
Theorem	nohalf 28487	An explicit expression for one half. This theorem avoids the axiom of infinity. (Contributed by Scott Fenton, 23-Jul-2025.)
⊢ ( 1s /su 2s) = ({ 0s } |s { 1s })
Theorem	expsval 28488	The value of surreal exponentiation. (Contributed by Scott Fenton, 24-Jul-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ ℤs) → (𝐴↑s𝐵) = if(𝐵 = 0s , 1s , if( 0s <s 𝐵, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))))))
Theorem	expnnsval 28489	Value of surreal exponentiation at a natural number. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁))
Theorem	exps0 28490	Surreal exponentiation to zero. (Contributed by Scott Fenton, 24-Jul-2025.)
⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
Theorem	exps1 28491	Surreal exponentiation to one. (Contributed by Scott Fenton, 24-Jul-2025.)
⊢ (𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴)
Theorem	expsp1 28492	Value of a surreal number raised to a non-negative integer power plus one. (Contributed by Scott Fenton, 6-Aug-2025.)
⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
Theorem	expscllem 28493*	Lemma for proving non-negative surreal integer exponentiation closure. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ 𝐹 ⊆ No     &   ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ·s 𝑦) ∈ 𝐹)    &   ⊢ 1s ∈ 𝐹    ⇒   ⊢ ((𝐴 ∈ 𝐹 ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ 𝐹)
Theorem	expscl 28494	Closure law for surreal exponentiation. (Contributed by Scott Fenton, 7-Aug-2025.)
⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ No )
Theorem	n0expscl 28495	Closure law for non-negative surreal integer exponentiation. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝐴 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ ℕ0s)
Theorem	nnexpscl 28496	Closure law for positive surreal integer exponentiation. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ ((𝐴 ∈ ℕs ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ ℕs)
Theorem	zexpscl 28497	Closure law for surreal integer exponentiation. (Contributed by Scott Fenton, 11-Dec-2025.)
⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ ℤs)
Theorem	expadds 28498	Sum of exponents law for surreals. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝐴 ∈ No ∧ 𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑀 +s 𝑁)) = ((𝐴↑s𝑀) ·s (𝐴↑s𝑁)))
Theorem	expsne0 28499	A non-negative surreal integer power is nonzero if its base is nonzero. (Contributed by Scott Fenton, 7-Aug-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s )
Theorem	expsgt0 28500	A non-negative surreal integer power is positive if its base is positive. (Contributed by Scott Fenton, 7-Aug-2025.)
⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁))