P. 283 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onscutleft 28201	A surreal ordinal is equal to the cut of its left set and the empty set. (Contributed by Scott Fenton, 29-Mar-2025.)
⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
Theorem	onscutlt 28202*	A surreal ordinal is the simplest number greater than all previous surreal ordinals. Theorem 15 of [Conway] p. 28. (Contributed by Scott Fenton, 4-Nov-2025.)
⊢ (𝐴 ∈ Ons → 𝐴 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅))
Theorem	bday11on 28203	The birthday function is one-to-one over the surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 = 𝐵)
Theorem	onnolt 28204	If a surreal ordinal is less than a given surreal, then it is simpler. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))
Theorem	onslt 28205	Less-than is the same as birthday comparison over surreal ordinals. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 <s 𝐵 ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
Theorem	onsiso 28206	The birthday function restricted to the surreal ordinals forms an order-preserving isomorphism with the regular ordinals. (Contributed by Scott Fenton, 8-Nov-2025.)
⊢ ( bday ↾ Ons) Isom <s , E (Ons, On)
Theorem	onswe 28207	Surreal less-than well-orders the surreal ordinals. Part of Theorem 15 of [Conway] p. 28. (Contributed by Scott Fenton, 6-Nov-2025.)
⊢ <s We Ons
Theorem	onsse 28208	Surreal less-than is set-like over the surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)
⊢ <s Se Ons
Theorem	onsis 28209*	Transfinite induction schema for surreal ordinals. (Contributed by Scott Fenton, 6-Nov-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ Ons → (∀𝑦 ∈ Ons (𝑦 <s 𝑥 → 𝜓) → 𝜑))    ⇒   ⊢ (𝐴 ∈ Ons → 𝜒)
Theorem	bdayon 28210*	The birthday of a surreal ordinal is the set of all previous ordinal birthdays. (Contributed by Scott Fenton, 7-Nov-2025.)
⊢ (𝐴 ∈ Ons → ( bday ‘𝐴) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}))
Theorem	onaddscl 28211	The surreal ordinals are closed under addition. (Contributed by Scott Fenton, 22-Aug-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) ∈ Ons)
Theorem	onmulscl 28212	The surreal ordinals are closed under multiplication. (Contributed by Scott Fenton, 22-Aug-2025.)
⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)
Theorem	peano2ons 28213	The successor of a surreal ordinal is a surreal ordinal. (Contributed by Scott Fenton, 22-Aug-2025.)
⊢ (𝐴 ∈ Ons → (𝐴 +s 1s ) ∈ Ons)
15.6.2  Surreal recursive sequences
Syntax	cseqs 28214	Extend class notation with the surreal recursive sequence builder.
class seqs𝑀( + , 𝐹)
Definition	df-seqs 28215*	Define a general-purpose sequence builder for surreal numbers. Compare df-seq 13911. 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 28216	Existence of the surreal sequence builder operation. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ seqs𝑀( + , 𝐹) ∈ V
Theorem	seqseq123d 28217	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 28218	Hypothesis builder for the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ Ⅎ𝑥𝑀    &   ⊢ Ⅎ𝑥 +    &   ⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥seqs𝑀( + , 𝐹)
Theorem	seqsval 28219*	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 28220	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 28221	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 28222*	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 28223*	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 28224*	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 28225	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 28226	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 28227	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 13856. (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 28228*	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 28229	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 28230*	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 28231*	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 28232*	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 28233*	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 28234*	An alternative definition of 𝐺 in terms of df-oi 9403. (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 28235*	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 28236*	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 28237*	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 28238*	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 28239*	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 28240	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 28241	The value of the surreal sequence bulder function at its initial value. (Contributed by Scott Fenton, 19-Apr-2025.)
⊢ (𝜑 → 𝑀 ∈ No )    ⇒   ⊢ (𝜑 → (seqs𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
Theorem	seqsp1 28242	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	cnn0s 28243	Declare the syntax for surreal non-negative integers.
class ℕ0s
Syntax	cnns 28244	Declare the syntax for surreal positive integers.
class ℕs
Definition	df-n0s 28245	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 12133. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
Definition	df-nns 28246	Define the set of positive surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs = (ℕ0s ∖ { 0s })
Theorem	n0sex 28247	The set of all non-negative surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s ∈ V
Theorem	nnsex 28248	The set of all positive surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs ∈ V
Theorem	peano5n0s 28249*	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 28250	The non-negative surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕ0s ⊆ No
Theorem	nnssn0s 28251	The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs ⊆ ℕ0s
Theorem	nnssno 28252	The positive surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ ℕs ⊆ No
Theorem	n0sno 28253	A non-negative surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
Theorem	nnsno 28254	A positive surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ∈ No )
Theorem	n0snod 28255	A non-negative surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕ0s)    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	nnsnod 28256	A positive surreal integer is a surreal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕs)    ⇒   ⊢ (𝜑 → 𝐴 ∈ No )
Theorem	nnn0s 28257	A positive surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)
Theorem	nnn0sd 28258	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 28259	Peano postulate: 0s is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
⊢ 0s ∈ ℕ0s
Theorem	peano2n0s 28260	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	dfn0s2 28261*	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 28262*	Principle of Mathematical Induction (inference schema). Compare nnind 12150 and finds 7832. (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	n0scut 28263	A cut form for non-negative surreal integers. (Contributed by Scott Fenton, 2-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))
Theorem	n0scut2 28264	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	n0ons 28265	A surreal natural is a surreal ordinal. (Contributed by Scott Fenton, 2-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
Theorem	nnne0s 28266	A surreal positive integer is non-zero. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )
Theorem	n0sge0 28267	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 28268	A positive integer is greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs → 0s <s 𝐴)
Theorem	elnns 28269	Membership in the positive surreal integers. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
Theorem	elnns2 28270	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 28271*	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 28272	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 28273	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 28274	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 28275	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 28276	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 28277	Surreal one is a non-negative surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ 1s ∈ ℕ0s
Theorem	1nns 28278	Surreal one is a positive surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ 1s ∈ ℕs
Theorem	peano2nns 28279	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 28280	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	n0sbday 28281	A non-negative surreal integer has a finite birthday. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)
Theorem	n0ssold 28282	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	n0sfincut 28283	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 28284	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	onltn0s 28285	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 28286*	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 28287	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 28288	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 28289	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 28290	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 28291	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	n0sltp1le 28292	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	n0sleltp1 28293	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	n0slem1lt 28294	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	bdayn0p1 28295	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 28296	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 28297	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 28298	Alternate definition of the positive surreal integers. Compare df-nn 12133. (Contributed by Scott Fenton, 6-Aug-2025.)
⊢ ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
Theorem	nnsind 28299*	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 28300	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))