P. 273 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27201-27300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nosupbnd1lem2 27201*	Lemma for nosupbnd1 27206. 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 27202*	Lemma for nosupbnd1 27206. 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 27203*	Lemma for nosupbnd1 27206. 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 27204*	Lemma for nosupbnd1 27206. 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 27205*	Lemma for nosupbnd1 27206. 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 27206*	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 27207*	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 27208*	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 27209*	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 27210*	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 27211*	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 27212*	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 27213*	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 27214*	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 27215*	Lemma for noinfbnd1 27221. 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 27216*	Lemma for noinfbnd1 27221. 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 27217*	Lemma for noinfbnd1 27221. 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 27218*	Lemma for noinfbnd1 27221. 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 27219*	Lemma for noinfbnd1 27221. 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 27220*	Lemma for noinfbnd1 27221. 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 27221*	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 27222*	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 27223*	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 27224*	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 27225*	Lemma for noeta 27235. 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 27226*	Lemma for noeta 27235. 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 27227*	Lemma for noeta 27235. 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 27228*	Lemma for noeta 27235. 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 27229*	Lemma for noeta 27235. 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 27230*	Lemma for noeta 27235. 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 27231*	Lemma for noeta 27235. 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 27232*	Lemma for noeta 27235. 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 27233*	Lemma for noeta 27235. 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 27234*	Lemma for noeta 27235. 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 27235*	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 27236	Declare the syntax for surreal less-than or equal.
class ≤s
Definition	df-sle 27237	Define the surreal less-than or equal predicate. Compare df-le 11250. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ ≤s = (( No × No ) ∖ ◡ <s )
Theorem	sltirr 27238	Surreal less-than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴)
Theorem	slttr 27239	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 27240	Surreal less-than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴))
Theorem	sltlin 27241	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 27242	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 27243	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 27244	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 27245	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 27246	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 27247	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 27248	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 27249	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 27250	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 27251	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 27252	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 27253	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 27254	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 27255	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 27256	Surreal trichotomy law. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ∨ 𝐵 ≤s 𝐴))
Theorem	maxs1 27257	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 27258	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 27259	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 27260	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 27261	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 27262	Surreal less-than implies not equal. (Contributed by Scott Fenton, 12-Mar-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴)
15.2.2  Birthday Theorems
Theorem	bdayfun 27263	The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ Fun bday
Theorem	bdayfn 27264	The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.)
⊢ bday Fn No
Theorem	bdaydm 27265	The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ dom bday = No
Theorem	bdayrn 27266	The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
⊢ ran bday = On
Theorem	bdayelon 27267	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 27268*	Lemma for nocvxmin 27269. 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 27269*	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 27270	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 27271	Declare the syntax for surreal set less-than.
class <<s
Definition	df-sslt 27272*	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 27273	Declare the syntax for the surreal cut operator.
class |s
Definition	df-scut 27274*	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 27275*	A version of noeta 27235 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 27276*	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 27277	The first argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
Theorem	ssltex2 27278	The second argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
Theorem	ssltss1 27279	The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
Theorem	ssltss2 27280	The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
Theorem	ssltsep 27281*	The separation property of surreal set less-than. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
Theorem	ssltd 27282*	Deduce surreal set less-than. (Contributed by Scott Fenton, 24-Sep-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ No )    &   ⊢ (𝜑 → 𝐵 ⊆ No )    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦)    ⇒   ⊢ (𝜑 → 𝐴 <<s 𝐵)
Theorem	ssltsepc 27283	Two elements of separated sets obey less-than. (Contributed by Scott Fenton, 20-Aug-2024.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌)
Theorem	ssltsepcd 27284	Two elements of separated sets obey less-than. Deduction form of ssltsepc 27283. (Contributed by Scott Fenton, 25-Sep-2024.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 <s 𝑌)
Theorem	sssslt1 27285	Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵)
Theorem	sssslt2 27286	Relation between surreal set less-than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶)
Theorem	nulsslt 27287	The empty set is less-than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
Theorem	nulssgt 27288	The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)
Theorem	conway 27289*	Conway's Simplicity Theorem. Given 𝐴 preceeding 𝐵, there is a unique surreal of minimal length separating them. This is a fundamental property of surreals and will be used (via surreal cuts) to prove many properties later on. Theorem from [Alling] p. 185. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
Theorem	scutval 27290*	The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
Theorem	scutcut 27291	Cut properties of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
Theorem	scutcl 27292	Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
Theorem	scutcld 27293	Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
⊢ (𝜑 → 𝐴 <<s 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 |s 𝐵) ∈ No )
Theorem	scutbday 27294*	The birthday of the surreal cut is equal to the minimum birthday in the gap. (Contributed by Scott Fenton, 8-Dec-2021.)
⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
Theorem	eqscut 27295*	Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
Theorem	eqscut2 27296*	Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑦)))))
Theorem	sslttr 27297	Transitive law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ∧ 𝐵 ≠ ∅) → 𝐴 <<s 𝐶)
Theorem	ssltun1 27298	Union law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶)
Theorem	ssltun2 27299	Union law for surreal set less-than. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶))
Theorem	scutun12 27300	Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (𝐴 |s 𝐵))