P. 277 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27601-27700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nodenselem6 27601*	The restriction of a surreal to the abstraction from nodenselem4 27599 is still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐴 ↾ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) ∈ No )
Theorem	nodenselem7 27602*	Lemma for nodense 27604. 𝐴 and 𝐵 are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 ∈ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} → (𝐴‘𝐶) = (𝐵‘𝐶)))
Theorem	nodenselem8 27603*	Lemma for nodense 27604. Give a condition for surreal less-than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 ↔ ((𝐴‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 1o ∧ (𝐵‘∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)}) = 2o)))
Theorem	nodense 27604*	Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Axiom SD of [Alling] p. 184. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑥 ∈ No (( bday ‘𝑥) ∈ ( bday ‘𝐴) ∧ 𝐴 <s 𝑥 ∧ 𝑥 <s 𝐵))
15.1.5  Full-Eta Property The theorems in this section are derived from "A clean way to separate sets of surreals" by Paolo Lipparini, https://doi.org/10.48550/arXiv.1712.03500.
Theorem	bdayimaon 27605	Lemma for full-eta properties. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.)
⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On)
Theorem	nolt02olem 27606	Lemma for nolt02o 27607. If 𝐴(𝑋) is undefined with 𝐴 surreal and 𝑋 ordinal, then dom 𝐴 ⊆ 𝑋. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋)
Theorem	nolt02o 27607	Given 𝐴 less-than 𝐵, equal to 𝐵 up to 𝑋, and undefined at 𝑋, then 𝐵(𝑋) = 2o. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → (𝐵‘𝑋) = 2o)
Theorem	nogt01o 27608	Given 𝐴 greater than 𝐵, equal to 𝐵 up to 𝑋, and 𝐵(𝑋) undefined, then 𝐴(𝑋) = 1o. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → (𝐴‘𝑋) = 1o)
Theorem	noresle 27609*	Restriction law for surreals. Lemma 2.1.4 of [Lipparini] p. 3. (Contributed by Scott Fenton, 5-Dec-2021.)
⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ (dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴 ∧ ∀𝑔 ∈ 𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)
Theorem	nomaxmo 27610*	A class of surreals has at most one maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
⊢ (𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦)
Theorem	nominmo 27611*	A class of surreals has at most one minimum. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑦 <s 𝑥)
Theorem	nosupprefixmo 27612*	In any class of surreals, there is at most one value of the prefix property. (Contributed by Scott Fenton, 26-Nov-2021.)
⊢ (𝐴 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥))
Theorem	noinfprefixmo 27613*	In any class of surreals, there is at most one value of the prefix property. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝐴 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥))
Theorem	nosupcbv 27614*	Lemma to change bound variables in a surreal supremum. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ 𝑆 = if(∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏, ((℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2o⟩}), (𝑐 ∈ {𝑑 ∣ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎))))
Theorem	nosupno 27615*	The next several theorems deal with a surreal "supremum". This surreal will ultimately be shown to bound 𝐴 below and bound the restriction of any surreal above. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 5-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) → 𝑆 ∈ No )
Theorem	nosupdm 27616*	The domain of the surreal supremum when there is no maximum. The primary point of this theorem is to change bound variable. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑧 ∣ ∃𝑝 ∈ 𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
Theorem	nosupbday 27617*	Birthday bounding law for surreal supremum. (Contributed by Scott Fenton, 5-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday “ 𝐴) ⊆ 𝑂)) → ( bday ‘𝑆) ⊆ 𝑂)
Theorem	nosupfv 27618*	The value of surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆‘𝐺) = (𝑈‘𝐺))
Theorem	nosupres 27619*	A restriction law for surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))
Theorem	nosupbnd1lem1 27620*	Lemma for nosupbnd1 27626. Establish a soft upper bound. (Contributed by Scott Fenton, 5-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
Theorem	nosupbnd1lem2 27621*	Lemma for nosupbnd1 27626. When there is no maximum, if any member of 𝐴 is a prolongment of 𝑆, then so are all elements of 𝐴 above it. (Contributed by Scott Fenton, 5-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) = 𝑆)
Theorem	nosupbnd1lem3 27622*	Lemma for nosupbnd1 27626. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not 2o. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2o)
Theorem	nosupbnd1lem4 27623*	Lemma for nosupbnd1 27626. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not undefined. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅)
Theorem	nosupbnd1lem5 27624*	Lemma for nosupbnd1 27626. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not 1o. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 1o)
Theorem	nosupbnd1lem6 27625*	Lemma for nosupbnd1 27626. Establish a hard upper bound when there is no maximum. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
Theorem	nosupbnd1 27626*	Bounding law from below for the surreal supremum. Proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
Theorem	nosupbnd2lem1 27627*	Bounding law from above when a set of surreals has a maximum. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2o⟩}))
Theorem	nosupbnd2 27628*	Bounding law from above for the surreal supremum. Proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ↔ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆))
Theorem	noinfcbv 27629*	Change bound variables for surreal infimum. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ 𝑇 = if(∃𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎, ((℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))))
Theorem	noinfno 27630*	The next several theorems deal with a surreal "infimum". This surreal will ultimately be shown to bound 𝐵 above and bound the restriction of any surreal below. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
Theorem	noinfdm 27631*	Next, we calculate the domain of 𝑇. This is mostly to change bound variables. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
Theorem	noinfbday 27632*	Birthday bounding law for surreal infimum. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ (((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday “ 𝐵) ⊆ 𝑂)) → ( bday ‘𝑇) ⊆ 𝑂)
Theorem	noinffv 27633*	The value of surreal infimum when there is no minimum. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑇‘𝐺) = (𝑈‘𝐺))
Theorem	noinfres 27634*	The restriction of surreal infimum when there is no minimum. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑇 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))
Theorem	noinfbnd1lem1 27635*	Lemma for noinfbnd1 27641. Establish a soft lower bound. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → ¬ (𝑈 ↾ dom 𝑇) <s 𝑇)
Theorem	noinfbnd1lem2 27636*	Lemma for noinfbnd1 27641. When there is no minimum, if any member of 𝐵 is a prolongment of 𝑇, then so are all elements below it. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑊 ↾ dom 𝑇) = 𝑇)
Theorem	noinfbnd1lem3 27637*	Lemma for noinfbnd1 27641. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not 1o. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 1o)
Theorem	noinfbnd1lem4 27638*	Lemma for noinfbnd1 27641. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not undefined. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
Theorem	noinfbnd1lem5 27639*	Lemma for noinfbnd1 27641. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not 2o. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 2o)
Theorem	noinfbnd1lem6 27640*	Lemma for noinfbnd1 27641. Establish a hard lower bound when there is no minimum. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
Theorem	noinfbnd1 27641*	Bounding law from above for the surreal infimum. Analagous to proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑈 ∈ 𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
Theorem	noinfbnd2lem1 27642*	Bounding law from below when a set of surreals has a minimum. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1o⟩}) <s (𝑍 ↾ suc dom 𝑈))
Theorem	noinfbnd2 27643*	Bounding law from below for the surreal infimum. Analagous to proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) → (∀𝑏 ∈ 𝐵 𝑍 <s 𝑏 ↔ ¬ 𝑇 <s (𝑍 ↾ dom 𝑇)))
Theorem	nosupinfsep 27644*	Given two sets of surreals, a surreal 𝑊 separates them iff its restriction to the maximum of dom 𝑆 and dom 𝑇 separates them. Corollary 4.4 of [Lipparini] p. 7. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))
Theorem	noetasuplem1 27645*	Lemma for noeta 27655. Establish that our final surreal really is a surreal. (Contributed by Scott Fenton, 6-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))    ⇒   ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No )
Theorem	noetasuplem2 27646*	Lemma for noeta 27655. The restriction of 𝑍 to dom 𝑆 is 𝑆. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))    ⇒   ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑍 ↾ dom 𝑆) = 𝑆)
Theorem	noetasuplem3 27647*	Lemma for noeta 27655. 𝑍 is an upper bound for 𝐴. Part of Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, 4-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))    ⇒   ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑋 ∈ 𝐴) → 𝑋 <s 𝑍)
Theorem	noetasuplem4 27648*	Lemma for noeta 27655. When 𝐴 and 𝐵 are separated, then 𝑍 is a lower bound for 𝐵. Part of Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, 7-Dec-2021.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))    ⇒   ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏)
Theorem	noetainflem1 27649*	Lemma for noeta 27655. Establish that this particular construction gives a surreal. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   ⊢ ((𝐴 ∈ V ∧ 𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑊 ∈ No )
Theorem	noetainflem2 27650*	Lemma for noeta 27655. The restriction of 𝑊 to the domain of 𝑇 is 𝑇. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)
Theorem	noetainflem3 27651*	Lemma for noeta 27655. 𝑊 bounds 𝐵 below . (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   ⊢ (((𝐴 ∈ V ∧ 𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑌 ∈ 𝐵) → 𝑊 <s 𝑌)
Theorem	noetainflem4 27652*	Lemma for noeta 27655. If 𝐴 precedes 𝐵, then 𝑊 is greater than 𝐴. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑊)
Theorem	noetalem1 27653*	Lemma for noeta 27655. Either 𝑆 or 𝑇 satisfies the final condition. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))    &   ⊢ 𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((𝑆 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏 ∧ ( bday ‘𝑆) ⊆ 𝑂)) ∨ (𝑇 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏 ∧ ( bday ‘𝑇) ⊆ 𝑂))))
Theorem	noetalem2 27654*	Lemma for noeta 27655. The full statement of the theorem with hypotheses in place. (Contributed by Scott Fenton, 10-Aug-2024.)
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑐 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂))
Theorem	noeta 27655*	The full-eta axiom for the surreal numbers. This is the single most important property of the surreals. It says that, given two sets of surreals such that one comes completely before the other, there is a surreal lying strictly between the two. Furthermore, if the birthdays of members of 𝐴 and 𝐵 are strictly bounded above by 𝑂, then 𝑂 non-strictly bounds the separator. Axiom FE of [Alling] p. 185. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday ‘𝑧) ⊆ 𝑂))
15.2  Initial consequences of Alling's axioms
15.2.1  Ordering Theorems
Syntax	csle 27656	Declare the syntax for surreal less-than or equal.
class ≤s
Definition	df-sle 27657	Define the surreal less-than or equal predicate. Compare df-le 11214. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ≤s = (( No × No ) ∖ ◡ <s )
Theorem	sltirr 27658	Surreal less-than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴)
Theorem	slttr 27659	Surreal less-than is transitive. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 <s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶))
Theorem	sltasym 27660	Surreal less-than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴))
Theorem	sltlin 27661	Surreal less-than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <s 𝐴))
Theorem	slttrieq2 27662	Trichotomy law for surreal less-than. (Contributed by Scott Fenton, 22-Apr-2012.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴)))
Theorem	slttrine 27663	Trichotomy law for surreals. (Contributed by Scott Fenton, 23-Nov-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≠ 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐵 <s 𝐴)))
Theorem	slenlt 27664	Surreal less-than or equal in terms of less-than. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
Theorem	sltnle 27665	Surreal less-than in terms of less-than or equal. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))
Theorem	sleloe 27666	Surreal less-than or equal in terms of less-than. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
Theorem	sletri3 27667	Trichotomy law for surreal less-than or equal. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴)))
Theorem	sltletr 27668	Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 <s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 <s 𝐶))
Theorem	slelttr 27669	Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶))
Theorem	sletr 27670	Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶))
Theorem	slttrd 27671	Surreal less-than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → 𝐵 <s 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 <s 𝐶)
Theorem	sltletrd 27672	Surreal less-than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    &   ⊢ (𝜑 → 𝐵 ≤s 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 <s 𝐶)
Theorem	slelttrd 27673	Surreal less-than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐴 ≤s 𝐵)    &   ⊢ (𝜑 → 𝐵 <s 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 <s 𝐶)
Theorem	sletrd 27674	Surreal less-than or equal is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐶 ∈ No )    &   ⊢ (𝜑 → 𝐴 ≤s 𝐵)    &   ⊢ (𝜑 → 𝐵 ≤s 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≤s 𝐶)
Theorem	slerflex 27675	Surreal less-than or equal is reflexive. Theorem 0(iii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴)
Theorem	sletric 27676	Surreal trichotomy law. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ∨ 𝐵 ≤s 𝐴))
Theorem	maxs1 27677	A surreal is less than or equal to the maximum of it and another. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ (𝐴 ∈ No → 𝐴 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))
Theorem	maxs2 27678	A surreal is less than or equal to the maximum of it and another. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))
Theorem	mins1 27679	The minimum of two surreals is less than or equal to the first. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴)
Theorem	mins2 27680	The minimum of two surreals is less than or equal to the second. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ (𝐵 ∈ No → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐵)
Theorem	sltled 27681	Surreal less-than implies less-than or equal. (Contributed by Scott Fenton, 16-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    &   ⊢ (𝜑 → 𝐴 <s 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≤s 𝐵)
Theorem	sltne 27682	Surreal less-than implies not equal. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴)
Theorem	sltlend 27683	Surreal less-than in terms of less-than or equal. (Contributed by Scott Fenton, 15-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ No )    &   ⊢ (𝜑 → 𝐵 ∈ No )    ⇒   ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≠ 𝐴)))
15.2.2  Birthday Theorems
Theorem	bdayfun 27684	The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ Fun bday
Theorem	bdayfn 27685	The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.)
⊢ bday Fn No
Theorem	bdaydm 27686	The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ dom bday = No
Theorem	bdayrn 27687	The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ ran bday = On
Theorem	bdayelon 27688	The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by Scott Fenton, 8-Dec-2021.)
⊢ ( bday ‘𝐴) ∈ On
Theorem	nocvxminlem 27689*	Lemma for nocvxmin 27690. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (( bday ‘𝑋) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑌) = ∩ ( bday “ 𝐴))) → ¬ 𝑋 <s 𝑌))
Theorem	nocvxmin 27690*	Given a nonempty convex class of surreals, there is a unique birthday-minimal element of that class. Lemma 0 of [Alling] p. 185. (Contributed by Scott Fenton, 30-Jun-2011.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∃!𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴))
Theorem	noprc 27691	The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ ¬ No ∈ V
15.3  Conway cut representation In [Conway] surreal numbers are represented as equivalence classes of cuts of previously defined surreal numbers. This is complicated to handle in ZFC without classes so we do not make it our definition. However, we can define a cut operator on surreals that behaves similarly. We introduce such an operator in this section and use it to define all surreals hearafter.
15.3.1  Conway cuts
Syntax	csslt 27692	Declare the syntax for surreal set less-than.
class <<s
Definition	df-sslt 27693*	Define the relation that holds iff one set of surreals completely precedes another. (Contributed by Scott Fenton, 7-Dec-2021.)
⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
Syntax	cscut 27694	Declare the syntax for the surreal cut operator.
class |s
Definition	df-scut 27695*	Define the cut operator on surreal numbers. This operator, which Conway takes as the primitive operator over surreals, picks the surreal lying between two sets of surreals of minimal birthday. Definition from [Gonshor] p. 7. (Contributed by Scott Fenton, 7-Dec-2021.)
⊢ |s = (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))
Theorem	noeta2 27696*	A version of noeta 27655 with fewer hypotheses but a weaker upper bound (Contributed by Scott Fenton, 7-Dec-2021.)
⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
Theorem	brsslt 27697*	Binary relation form of the surreal set less-than relation. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
Theorem	ssltex1 27698	The first argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
Theorem	ssltex2 27699	The second argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
Theorem	ssltss1 27700	The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )