P. 284 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	seqsfn 28301	The surreal sequence builder is a function. (Contributed by Scott Fenton, 19-Apr-2025.)
⊢ (𝜑 → 𝑀 ∈ No )    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))    ⇒   ⊢ (𝜑 → seqs𝑀( + , 𝐹) Fn 𝑍)
Theorem	seqs1 28302	The value of the surreal sequence builder function at its initial value. (Contributed by Scott Fenton, 19-Apr-2025.)
⊢ (𝜑 → 𝑀 ∈ No )    ⇒   ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
Theorem	seqsp1 28303	The value of the surreal sequence builder at a successor. (Contributed by Scott Fenton, 19-Apr-2025.)
⊢ (𝜑 → 𝑀 ∈ No )    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    ⇒   ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))
15.6.3  Natural numbers
Syntax	cn0s 28304	Declare the syntax for surreal non-negative integers.
class ℕ0s
Syntax	cnns 28305	Declare the syntax for surreal positive integers.
class ℕs
Definition	df-n0s 28306	Define the set of non-negative surreal integers. This set behaves similarly to ω and ℕ0, but it is a set of surreal numbers. Like those two sets, it satisfies the Peano axioms and is closed under (surreal) addition and multiplication. Compare df-nn 12175. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
Definition	df-nns 28307	Define the set of positive surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs = (ℕ0s ∖ { 0s })
Theorem	n0sexg 28308	The set of all non-negative surreal integers exists. This theorem avoids the axiom of infinity by including it as an antecedent. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (ω ∈ V → ℕ0s ∈ V)
Theorem	n0sex 28309	The set of all non-negative surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s ∈ V
Theorem	nnsex 28310	The set of all positive surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs ∈ V
Theorem	peano5n0s 28311*	Peano's inductive postulate for non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → ℕ0s ⊆ 𝐴)
Theorem	n0ssno 28312	The non-negative surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s ⊆ No
Theorem	nnssn0s 28313	The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs ⊆ ℕ0s
Theorem	nnssno 28314	The positive surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs ⊆ No
Theorem	n0no 28315	A non-negative surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
Theorem	nnno 28316	A positive surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ∈ No )
Theorem	n0nod 28317	A non-negative surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	nnnod 28318	A positive surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕs)    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	nnn0s 28319	A positive surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)
Theorem	nnn0sd 28320	A positive surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕs)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℕ0s)
Theorem	0n0s 28321	Peano postulate: 0s is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ 0s ∈ ℕ0s
Theorem	peano2n0s 28322	Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)
Theorem	peano2n0sd 28323	Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. Deduction form. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → (𝐴 +s 1s ) ∈ ℕ0s)
Theorem	dfn0s2 28324*	Alternate definition of the set of non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s = ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
Theorem	n0sind 28325*	Principle of Mathematical Induction (inference schema). Compare nnind 12192 and finds 7847. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ (𝑥 = 0s → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 +s 1s ) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕ0s → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ℕ0s → 𝜏)
Theorem	n0cut 28326	A cut form for non-negative surreal integers. (Contributed by Scott Fenton, 2-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))
Theorem	n0cut2 28327	A cut form for the successor of a non-negative surreal integer. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) = ({𝐴} |s ∅))
Theorem	n0on 28328	A surreal natural is a surreal ordinal. (Contributed by Scott Fenton, 2-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
Theorem	nnne0s 28329	A surreal positive integer is nonzero. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )
Theorem	n0sge0 28330	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 28331	A positive integer is greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 0s <s 𝐴)
Theorem	elnns 28332	Membership in the positive surreal integers. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
Theorem	elnns2 28333	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 28334*	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 28335	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 28336	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 28337	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 28338	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 28339	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 28340	Surreal one is a non-negative surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ 1s ∈ ℕ0s
Theorem	1nns 28341	Surreal one is a positive surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ 1s ∈ ℕs
Theorem	peano2nns 28342	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 28343	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 28344	A non-negative surreal integer has a finite birthday. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)
Theorem	n0ssoldg 28345	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 28346	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 28347	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 28348	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 28349	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 28350	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 28351*	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 28352	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 28353	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 28354	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 28355	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 28356	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 28357	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 28358	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 28359	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 28360	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 28361	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 28362	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 28363	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 28364	Alternate definition of the positive surreal integers. Compare df-nn 12175. (Contributed by Scott Fenton, 6-Aug-2025.)
⊢ ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
Theorem	nnsind 28365*	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 28366	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 28367	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 28368*	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 28369	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 28370	Declare the syntax for surreal integers.
class ℤs
Definition	df-zs 28371	Define the surreal integers. Compare dfz2 12543. (Contributed by Scott Fenton, 17-May-2025.)
⊢ ℤs = ( -s “ (ℕs × ℕs))
Theorem	zsex 28372	The surreal integers form a set. (Contributed by Scott Fenton, 17-May-2025.)
⊢ ℤs ∈ V
Theorem	zssno 28373	The surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-May-2025.)
⊢ ℤs ⊆ No
Theorem	zno 28374	A surreal integer is a surreal. (Contributed by Scott Fenton, 17-May-2025.)
⊢ (𝐴 ∈ ℤs → 𝐴 ∈ No )
Theorem	znod 28375	A surreal integer is a surreal. Deduction form. (Contributed by Scott Fenton, 17-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℤs)    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	elzs 28376*	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 28377	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 28378	A positive surreal integer is a surreal integer. (Contributed by Scott Fenton, 17-May-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℤs)
Theorem	nnzsd 28379	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 28380	Zero is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ 0s ∈ ℤs
Theorem	n0zs 28381	A non-negative surreal integer is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℤs)
Theorem	n0zsd 28382	A non-negative surreal integer is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤs)
Theorem	1zs 28383	One is a surreal integer. (Contributed by Scott Fenton, 24-Jul-2025.)
⊢ 1s ∈ ℤs
Theorem	znegscl 28384	The surreal integers are closed under negation. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℤs → ( -us ‘𝐴) ∈ ℤs)
Theorem	znegscld 28385	The surreal integers are closed under negation. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝜑 → 𝐴 ∈ ℤs)    ⇒   ⊢ (𝜑 → ( -us ‘𝐴) ∈ ℤs)
Theorem	zaddscl 28386	The surreal integers are closed under addition. (Contributed by Scott Fenton, 25-Jul-2025.)
⊢ ((𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs) → (𝐴 +s 𝐵) ∈ ℤs)
Theorem	zaddscld 28387	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 28388	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 28389	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 28390	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 28391	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 28392	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 28393	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 28394	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 28395	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 28396*	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 28397*	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 28398	A surreal integer has a finite birthday. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℤs → ( bday ‘𝐴) ∈ ω)
Theorem	zcuts 28399	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 28400	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) = ∅))