P. 287 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28601-28700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onsbnd 28601	The surreals of a given birthday are bounded above by that ordinal. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ≤s 𝐴)
Theorem	onsbnd2 28602	The surreals of a given birthday are bounded below by the negative of that ordinal. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -us ‘𝐴) ≤s 𝐵)
15.6.2  Surreal recursive sequences
Syntax	cseqs 28603	Extend class notation with the surreal recursive sequence builder.
class seqs𝑀( + , 𝐹)
Definition	df-seqs 28604*	Define a general-purpose sequence builder for surreal numbers. Compare df-seq 14114. Note that in the theorems we develop here, we do not require 𝑀 to be an integer. This is because there are infinite surreal numbers and we may want to start our sequence there. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
Theorem	seqsex 28605	Existence of the surreal sequence builder operation. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ seqs𝑀( + , 𝐹) ∈ V
Theorem	seqseq123d 28606	Equality deduction for the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝑀 = 𝑁)    &   ⊢ (𝜑 → + = 𝑄)    &   ⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺))
Theorem	nfseqs 28607	Hypothesis builder for the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ Ⅎ𝑥𝑀    &   ⊢ Ⅎ𝑥 +    &   ⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥seqs𝑀( + , 𝐹)
Theorem	seqsval 28608*	The value of the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω))    ⇒   ⊢ (𝜑 → seqs𝑀( + , 𝐹) = ran 𝑅)
Theorem	noseqex 28609	The next several theorems develop the concept of a countable sequence of surreals. This set is denoted by 𝑍 here and is the analogue of the upper integer sets for surreal numbers. However, we do not require the starting point to be an integer so we can accommodate infinite numbers. This first theorem establishes that 𝑍 is a set. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    ⇒   ⊢ (𝜑 → 𝑍 ∈ V)
Theorem	noseq0 28610	The surreal 𝐴 is a member of the sequence starting at 𝐴. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝑍)
Theorem	noseqp1 28611*	One plus an element of 𝑍 is an element of 𝑍. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ 𝑍)    ⇒   ⊢ (𝜑 → (𝐵 +s 1s ) ∈ 𝑍)
Theorem	noseqind 28612*	Peano's inductive postulate for surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 +s 1s ) ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑍 ⊆ 𝐵)
Theorem	noseqinds 28613*	Induction schema for surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑦 = (𝑧 +s 1s ) → (𝜓 ↔ 𝜏))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜂))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝜃 → 𝜏))    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝜂)
Theorem	noseqssno 28614	A surreal sequence is a subset of the surreals. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ No )    ⇒   ⊢ (𝜑 → 𝑍 ⊆ No )
Theorem	noseqno 28615	An element of a surreal sequence is a surreal. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐵 ∈ No )
Theorem	om2noseq0 28616	The mapping 𝐺 is a one-to-one mapping from ω onto a countable sequence of surreals that will be used to show the properties of seqs. This theorem shows the value of 𝐺 at ordinal zero. Compare the series of theorems starting at om2uz0i 14059. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    ⇒   ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
Theorem	om2noseqsuc 28617*	The value of 𝐺 at a successor. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝐴 ∈ ω)    ⇒   ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) +s 1s ))
Theorem	om2noseqfo 28618	Function statement for 𝐺. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   ⊢ (𝜑 → 𝐺:ω–onto→𝑍)
Theorem	om2noseqlt 28619*	Surreal less-than relation for 𝐺. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) <s (𝐺‘𝐵)))
Theorem	om2noseqlt2 28620*	The mapping 𝐺 preserves order. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) <s (𝐺‘𝐵)))
Theorem	om2noseqf1o 28621*	𝐺 is a bijection. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   ⊢ (𝜑 → 𝐺:ω–1-1-onto→𝑍)
Theorem	om2noseqiso 28622*	𝐺 is an isomorphism from the finite ordinals to a surreal sequence. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   ⊢ (𝜑 → 𝐺 Isom E , <s (ω, 𝑍))
Theorem	om2noseqoi 28623*	An alternative definition of 𝐺 in terms of df-oi 9482. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   ⊢ (𝜑 → 𝐺 = OrdIso( <s , 𝑍))
Theorem	om2noseqrdg 28624*	A helper lemma for the value of a recursive definition generator on a surreal sequence with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ ω) → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)
Theorem	noseqrdglem 28625*	A helper lemma for the value of a recursive defintion generator on surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
Theorem	noseqrdgfn 28626*	The recursive definition generator on surreal sequences is a function. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    &   ⊢ (𝜑 → 𝑆 = ran 𝑅)    ⇒   ⊢ (𝜑 → 𝑆 Fn 𝑍)
Theorem	noseqrdg0 28627*	Initial value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    &   ⊢ (𝜑 → 𝑆 = ran 𝑅)    ⇒   ⊢ (𝜑 → (𝑆‘𝐶) = 𝐴)
Theorem	noseqrdgsuc 28628*	Successor value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 19-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    &   ⊢ (𝜑 → 𝑆 = ran 𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆‘𝐵)))
Theorem	seqsfn 28629	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 28630	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 28631	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 28632	Declare the syntax for surreal non-negative integers.
class ℕ0s
Syntax	cnns 28633	Declare the syntax for surreal positive integers.
class ℕs
Definition	df-n0s 28634	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 12306. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
Definition	df-nns 28635	Define the set of positive surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs = (ℕ0s ∖ { 0s })
Theorem	n0sexg 28636	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 28637	The set of all non-negative surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s ∈ V
Theorem	nnsex 28638	The set of all positive surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs ∈ V
Theorem	peano5n0s 28639*	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 28640	The non-negative surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s ⊆ No
Theorem	nnssn0s 28641	The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs ⊆ ℕ0s
Theorem	nnssno 28642	The positive surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs ⊆ No
Theorem	n0no 28643	A non-negative surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
Theorem	nnno 28644	A positive surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ∈ No )
Theorem	n0nod 28645	A non-negative surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	nnnod 28646	A positive surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕs)    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	nnn0s 28647	A positive surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)
Theorem	nnn0sd 28648	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 28649	Peano postulate: 0s is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ 0s ∈ ℕ0s
Theorem	peano2n0s 28650	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 28651	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 28652*	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 28653*	Principle of Mathematical Induction (inference schema). Compare nnind 12323 and finds 7891. (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 28654	A cut form for non-negative surreal integers. (Contributed by Scott Fenton, 2-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))
Theorem	n0cut2 28655	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 28656	A surreal natural is a surreal ordinal. (Contributed by Scott Fenton, 2-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
Theorem	nnne0s 28657	A surreal positive integer is nonzero. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )
Theorem	n0sge0 28658	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 28659	A positive integer is greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 0s <s 𝐴)
Theorem	elnns 28660	Membership in the positive surreal integers. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
Theorem	elnns2 28661	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 28662*	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 28663	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 28664	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 28665	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 28666	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 28667	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 28668	Surreal one is a non-negative surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ 1s ∈ ℕ0s
Theorem	1nns 28669	Surreal one is a positive surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ 1s ∈ ℕs
Theorem	peano2nns 28670	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 28671	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 28672	A non-negative surreal integer has a finite birthday. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)
Theorem	n0ssoldg 28673	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 28674	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 28675	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 28676	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 28677	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 28678	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 28679*	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 28680	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 28681	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 28682	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 28683	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 28684	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 28685	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 28686	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 28687	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 28688	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 28689	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 28690	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 28691	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 28692	Alternate definition of the positive surreal integers. Compare df-nn 12306. (Contributed by Scott Fenton, 6-Aug-2025.)
⊢ ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
Theorem	nnsind 28693*	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 28694	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 28695	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 28696*	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 28697	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 28698	Declare the syntax for surreal integers.
class ℤs
Definition	df-zs 28699	Define the surreal integers. Compare dfz2 12682. (Contributed by Scott Fenton, 17-May-2025.)
⊢ ℤs = ( -s “ (ℕs × ℕs))
Theorem	zsex 28700	The surreal integers form a set. (Contributed by Scott Fenton, 17-May-2025.)
⊢ ℤs ∈ V