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Theorem List for Metamath Proof Explorer - 33101-33200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnosupno 33101* 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(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((𝐴 No 𝐴𝑉) → 𝑆 No )
 
Theoremnosupdm 33102* 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(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑧 ∣ ∃𝑝𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐴𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
 
Theoremnosupbday 33103* Birthday bounding law for surreal supremum. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) ⊆ suc ( bday 𝐴))
 
Theoremnosupfv 33104* The value of surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴𝐺 ∈ dom 𝑈 ∧ ∀𝑣𝐴𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆𝐺) = (𝑈𝐺))
 
Theoremnosupres 33105* A restriction law for surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴𝐺 ∈ dom 𝑈 ∧ ∀𝑣𝐴𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))
 
Theoremnosupbnd1lem1 33106* Lemma for nosupbnd1 33112. Establish a soft upper bound. (Contributed by Scott Fenton, 5-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
 
Theoremnosupbnd1lem2 33107* Lemma for nosupbnd1 33112. 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(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) = 𝑆)
 
Theoremnosupbnd1lem3 33108* Lemma for nosupbnd1 33112. 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(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2o)
 
Theoremnosupbnd1lem4 33109* Lemma for nosupbnd1 33112. 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(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅)
 
Theoremnosupbnd1lem5 33110* Lemma for nosupbnd1 33112. 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(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 1o)
 
Theoremnosupbnd1lem6 33111* Lemma for nosupbnd1 33112. Establish a hard upper bound when there is no maximum. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
 
Theoremnosupbnd1 33112* Bounding law from below for the surreal supremum. Proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
 
Theoremnosupbnd2lem1 33113* Bounding law from above when a set of surreals has a maximum. (Contributed by Scott Fenton, 6-Dec-2021.)
(((𝑈𝐴 ∧ ∀𝑦𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑍 No ) ∧ ∀𝑎𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2o⟩}))
 
Theoremnosupbnd2 33114* Bounding law from above for the surreal supremum. Proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))       ((𝐴 No 𝐴 ∈ V ∧ 𝑍 No ) → (∀𝑎𝐴 𝑎 <s 𝑍 ↔ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆))
 
Theoremnoetalem1 33115* Lemma for noeta 33120. Establish that our final surreal really is a surreal. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )
 
Theoremnoetalem2 33116* Lemma for noeta 33120. 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(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       (((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑋𝐴) → 𝑋 <s 𝑍)
 
Theoremnoetalem3 33117* Lemma for noeta 33120. 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(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∀𝑏𝐵 𝑍 <s 𝑏)
 
Theoremnoetalem4 33118* Lemma for noeta 33120. Bound the birthday of 𝑍 above. (Contributed by Scott Fenton, 6-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))
 
Theoremnoetalem5 33119* Lemma for noeta 33120. The full statement of the theorem with hypotheses. (Contributed by Scott Fenton, 7-Dec-2021.)
𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))       (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∃𝑧 No (∀𝑎𝐴 𝑎 <s 𝑧 ∧ ∀𝑏𝐵 𝑧 <s 𝑏 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
 
Theoremnoeta 33120* 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, there is an upper bound on the birthday of that surreal. Alling's axiom FE. (Contributed by Scott Fenton, 7-Dec-2021.)
(((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐴 𝑥 <s 𝑧 ∧ ∀𝑦𝐵 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
 
20.9.26  Surreal numbers - ordering theorems
 
Syntaxcsle 33121 Declare the syntax for surreal less than or equal.
class ≤s
 
Definitiondf-sle 33122 Define the surreal less than or equal predicate. Compare df-le 10670. (Contributed by Scott Fenton, 8-Dec-2021.)
≤s = (( No × No ) ∖ <s )
 
Theoremsltirr 33123 Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → ¬ 𝐴 <s 𝐴)
 
Theoremslttr 33124 Surreal less than is transitive. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐶) → 𝐴 <s 𝐶))
 
Theoremsltasym 33125 Surreal less than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴))
 
Theoremsltlin 33126 Surreal less than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐴 = 𝐵𝐵 <s 𝐴))
 
Theoremslttrieq2 33127 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)))
 
Theoremslttrine 33128 Trichotomy law for surreals. (Contributed by Scott Fenton, 23-Nov-2021.)
((𝐴 No 𝐵 No ) → (𝐴𝐵 ↔ (𝐴 <s 𝐵𝐵 <s 𝐴)))
 
Theoremslenlt 33129 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))
 
Theoremsltnle 33130 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))
 
Theoremsleloe 33131 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 𝐵𝐴 = 𝐵)))
 
Theoremsletri3 33132 Trichotomy law for surreal less than or equal. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No ) → (𝐴 = 𝐵 ↔ (𝐴 ≤s 𝐵𝐵 ≤s 𝐴)))
 
Theoremsltletr 33133 Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 <s 𝐵𝐵 ≤s 𝐶) → 𝐴 <s 𝐶))
 
Theoremslelttr 33134 Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 ≤s 𝐵𝐵 <s 𝐶) → 𝐴 <s 𝐶))
 
Theoremsletr 33135 Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 ≤s 𝐵𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶))
 
Theoremslttrd 33136 Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝜑𝐴 No )    &   (𝜑𝐵 No )    &   (𝜑𝐶 No )    &   (𝜑𝐴 <s 𝐵)    &   (𝜑𝐵 <s 𝐶)       (𝜑𝐴 <s 𝐶)
 
Theoremsltletrd 33137 Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝜑𝐴 No )    &   (𝜑𝐵 No )    &   (𝜑𝐶 No )    &   (𝜑𝐴 <s 𝐵)    &   (𝜑𝐵 ≤s 𝐶)       (𝜑𝐴 <s 𝐶)
 
Theoremslelttrd 33138 Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝜑𝐴 No )    &   (𝜑𝐵 No )    &   (𝜑𝐶 No )    &   (𝜑𝐴 ≤s 𝐵)    &   (𝜑𝐵 <s 𝐶)       (𝜑𝐴 <s 𝐶)
 
Theoremsletrd 33139 Surreal less than or equal is transitive. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝜑𝐴 No )    &   (𝜑𝐵 No )    &   (𝜑𝐶 No )    &   (𝜑𝐴 ≤s 𝐵)    &   (𝜑𝐵 ≤s 𝐶)       (𝜑𝐴 ≤s 𝐶)
 
20.9.27  Surreal numbers - birthday theorems
 
Theorembdayfun 33140 The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
Fun bday
 
Theorembdayfn 33141 The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.)
bday Fn No
 
Theorembdaydm 33142 The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.)
dom bday = No
 
Theorembdayrn 33143 The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
ran bday = On
 
Theorembdayelon 33144 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
 
Theoremnocvxminlem 33145* Lemma for nocvxmin 33146. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ¬ 𝑋 <s 𝑌))
 
Theoremnocvxmin 33146* Given a nonempty convex class of surreals, there is a unique birthday-minimal element of that class. (Contributed by Scott Fenton, 30-Jun-2011.)
((𝐴 ≠ ∅ ∧ 𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∃!𝑤𝐴 ( bday 𝑤) = ( bday 𝐴))
 
Theoremnoprc 33147 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ No ∈ V
 
20.9.28  Surreal numbers: Conway cuts
 
Syntaxcsslt 33148 Declare the syntax for surreal set less than.
class <<s
 
Definitiondf-sslt 33149* Define the relationship that holds iff one set of surreals completely precedes another. (Contributed by Scott Fenton, 7-Dec-2021.)
<<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}
 
Syntaxcscut 33150 Declare the syntax for the surreal cut operator.
class |s
 
Definitiondf-scut 33151* 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. (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)})))
 
Theorembrsslt 33152* 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 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))
 
Theoremssltex1 33153 The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 ∈ V)
 
Theoremssltex2 33154 The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 ∈ V)
 
Theoremssltss1 33155 The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 No )
 
Theoremssltss2 33156 The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 No )
 
Theoremssltsep 33157* The separation property of surreal set less than. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)
 
Theoremsssslt1 33158 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)
 
Theoremsssslt2 33159 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)
 
Theoremnulsslt 33160 The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)
 
Theoremnulssgt 33161 The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No 𝐴 <<s ∅)
 
Theoremconway 33162* 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. (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)}))
 
Theoremscutval 33163* 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)})))
 
Theoremscutcut 33164 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))
 
Theoremscutbday 33165* 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)}))
 
Theoremsslttr 33166 Transitive law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐵 <<s 𝐶𝐵 ≠ ∅) → 𝐴 <<s 𝐶)
 
Theoremssltun1 33167 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)
 
Theoremssltun2 33168 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))
 
Theoremscutun12 33169 Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴𝐶) |s (𝐵𝐷)) = (𝐴 |s 𝐵))
 
Theoremdmscut 33170 The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
dom |s = <<s
 
Theoremscutf 33171 Functionhood statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
|s : <<s ⟶ No
 
Theoremetasslt 33172* A restatement of noeta 33120 using set less than. (Contributed by Scott Fenton, 10-Dec-2021.)
(𝐴 <<s 𝐵 → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))
 
Theoremscutbdaybnd 33173 An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.)
(𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))
 
Theoremscutbdaylt 33174 If a surreal lies in a gap and is not equal to the cut, its birthday is greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.)
((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday 𝑋))
 
Theoremslerec 33175* A comparison law for surreals considered as cuts of sets of surreals. In Conway's treatment, this is the definition of less than or equal. (Contributed by Scott Fenton, 11-Dec-2021.)
(((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑𝐷 𝑋 <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s 𝑌)))
 
Theoremsltrec 33176* A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)
(((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌)))
 
20.9.29  Surreal numbers - cuts and options
 
Syntaxcmade 33177 Declare the symbol for the made by function.
class M
 
Syntaxcold 33178 Declare the symbol for the older than function.
class O
 
Syntaxcnew 33179 Declare the symbol for the new on function.
class N
 
Syntaxcleft 33180 Declare the symbol for the left option function.
class L
 
Syntaxcright 33181 Declare the symbol for the right option function.
class R
 
Definitiondf-made 33182 Define the made by function. This function carries an ordinal to all surreals made by sections of surreals older than it. (Contributed by Scott Fenton, 17-Dec-2021.)
M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ran 𝑓 × 𝒫 ran 𝑓))))
 
Definitiondf-old 33183 Define the older than function. This function carries an ordinal to all surreals made by a previous ordinal. (Contributed by Scott Fenton, 17-Dec-2021.)
O = (𝑥 ∈ On ↦ ( M “ 𝑥))
 
Definitiondf-new 33184 Define the newer than function. This function carries an ordinal to all surreals made on that day. (Contributed by Scott Fenton, 17-Dec-2021.)
N = (𝑥 ∈ On ↦ (( O ‘𝑥) ∖ ( M ‘𝑥)))
 
Definitiondf-left 33185* Define the left options of a surreal. This is the set of surreals that are "closest" on the left to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.)
L = (𝑥 No ↦ {𝑦 ∈ ( O ‘( bday 𝑥)) ∣ ∀𝑧 No ((𝑦 <s 𝑧𝑧 <s 𝑥) → ( bday 𝑦) ∈ ( bday 𝑧))})
 
Definitiondf-right 33186* Define the left options of a surreal. This is the set of surreals that are "closest" on the right to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.)
R = (𝑥 No ↦ {𝑦 ∈ ( O ‘( bday 𝑥)) ∣ ∀𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → ( bday 𝑦) ∈ ( bday 𝑧))})
 
Theoremmadeval 33187 The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
(𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
 
Theoremmadeval2 33188* Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
(𝐴 ∈ On → ( M ‘𝐴) = {𝑥 No ∣ ∃𝑎 ∈ 𝒫 ( M “ 𝐴)∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
 
20.9.30  Quantifier-free definitions
 
Syntaxctxp 33189 Declare the syntax for tail Cartesian product.
class (𝐴𝐵)
 
Syntaxcpprod 33190 Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)
 
Syntaxcsset 33191 Declare the subset relationship class.
class SSet
 
Syntaxctrans 33192 Declare the transitive set class.
class Trans
 
Syntaxcbigcup 33193 Declare the set union relationship.
class Bigcup
 
Syntaxcfix 33194 Declare the syntax for the fixpoints of a class.
class Fix 𝐴
 
Syntaxclimits 33195 Declare the class of limit ordinals.
class Limits
 
Syntaxcfuns 33196 Declare the syntax for the class of all function.
class Funs
 
Syntaxcsingle 33197 Declare the syntax for the singleton function.
class Singleton
 
Syntaxcsingles 33198 Declare the syntax for the class of all singletons.
class Singletons
 
Syntaxcimage 33199 Declare the syntax for the image functor.
class Image𝐴
 
Syntaxccart 33200 Declare the syntax for the cartesian function.
class Cart
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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