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Theorem List for Metamath Proof Explorer - 31001-31100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-wsucOLD 31001 Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) Obsolete version of df-wsuc 31000 as of 10-Oct-2021. (New usage is discouraged.)
wsucOLD(𝑅, 𝐴, 𝑋) = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)

Definitiondf-wlim 31002* Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}

Definitiondf-wlimOLD 31003* Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) Obsolete version of df-wlim 31002 as of 10-Oct-2021. (New usage is discouraged.)
WLimOLD(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ sup(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}

Theoremwsuceq123 31004 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))

Theoremwsuceq1 31005 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 = 𝑆 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋))

Theoremwsuceq2 31006 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))

Theoremwsuceq3 31007 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))

Theoremnfwsuc 31008 Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
𝑥𝑅    &   𝑥𝐴    &   𝑥𝑋       𝑥wsuc(𝑅, 𝐴, 𝑋)

Theoremwlimeq12 31009 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
((𝑅 = 𝑆𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))

Theoremwlimeq1 31010 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
(𝑅 = 𝑆 → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐴))

Theoremwlimeq2 31011 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
(𝐴 = 𝐵 → WLim(𝑅, 𝐴) = WLim(𝑅, 𝐵))

Theoremnfwlim 31012 Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
𝑥𝑅    &   𝑥𝐴       𝑥WLim(𝑅, 𝐴)

Theoremelwlim 31013 Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
(𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋𝐴𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))

TheoremelwlimOLD 31014 Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) Obsolete version of elwlim 31013 as of 10-Oct-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(𝑋 ∈ WLimOLD(𝑅, 𝐴) ↔ (𝑋𝐴𝑋 ≠ sup(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))

Theoremwzel 31015 The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
((𝑅 We 𝐴𝑅 Se 𝐴𝐴 ≠ ∅) → inf(𝐴, 𝐴, 𝑅) ∈ 𝐴)

TheoremwzelOLD 31016 The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) Obsolete version of wzel 31015 as of 10-Oct-2021. (New usage is discouraged.) (Proof modification is discouraged.)
((𝑅 We 𝐴𝑅 Se 𝐴𝐴 ≠ ∅) → sup(𝐴, 𝐴, 𝑅) ∈ 𝐴)

Theoremwsuclem 31017* Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑 → ∃𝑤𝐴 𝑋𝑅𝑤)       (𝜑 → ∃𝑥𝐴 (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))

TheoremwsuclemOLD 31018* Obsolete version of wsuclem 31017 as of 10-Oct-2021. (Contributed by Scott Fenton, 15-Jun-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑 → ∃𝑤𝐴 𝑋𝑅𝑤)       (𝜑 → ∃𝑥𝐴 (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ Pred (𝑅, 𝐴, 𝑋)𝑦𝑅𝑧)))

Theoremwsucex 31019 Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
(𝜑𝑅 Or 𝐴)       (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

Theoremwsuccl 31020* If 𝑋 is a set with an 𝑅 successor in 𝐴, then its well-founded successor is a member of 𝐴. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑 → ∃𝑦𝐴 𝑋𝑅𝑦)       (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)

Theoremwsuclb 31021 A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝐴)    &   (𝜑𝑋𝑅𝑌)       (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))

Theoremwlimss 31022 The class of limit points is a subclass of the base class. (Contributed by Scott Fenton, 16-Jun-2018.)
WLim(𝑅, 𝐴) ⊆ 𝐴

21.8.20  Founded Recursion

Theoremfrr3g 31023* Functions defined by founded recursion are identical up to relation, domain, and characteristic function. General version of frr3. (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)

Theoremfrrlem1 31024* Lemma for founded recursion. The final item we are interested in is the union of acceptable functions 𝐵. This lemma just changes bound variables for later use. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))}

Theoremfrrlem2 31025* Lemma for founded recursion. An acceptable function is a function. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝑔𝐵 → Fun 𝑔)

Theoremfrrlem3 31026* Lemma for founded recursion. An acceptable function's domain is a subset of 𝐴. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝑔𝐵 → dom 𝑔𝐴)

Theoremfrrlem4 31027* Lemma for founded recursion. Properties of the restriction of an acceptable function to the domain of another acceptable function. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       ((𝑔𝐵𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))

Theoremfrrlem5 31028* Lemma for founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       ((𝑔𝐵𝐵) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))

Theoremfrrlem5b 31029* Lemma for founded recursion. The union of a subclass of 𝐵 is a relationship. (Contributed by Paul Chapman, 29-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝐶𝐵 → Rel 𝐶)

Theoremfrrlem5c 31030* Lemma for founded recursion. The union of a subclass of 𝐵 is a function. (Contributed by Paul Chapman, 29-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝐶𝐵 → Fun 𝐶)

Theoremfrrlem5d 31031* Lemma for founded recursion. The domain of the union of a subset of 𝐵 is a subset of 𝐴. (Contributed by Paul Chapman, 29-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝐶𝐵 → dom 𝐶𝐴)

Theoremfrrlem5e 31032* Lemma for founded recursion. The domain of the union of a subset of 𝐵 is closed under predecessors. (Contributed by Paul Chapman, 1-May-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝐶𝐵 → (𝑋 ∈ dom 𝐶 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))

Theoremfrrlem6 31033* Lemma for founded recursion. The union of all acceptable functions is a relationship. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}    &   𝐹 = 𝐵       Rel 𝐹

Theoremfrrlem7 31034* Lemma for founded recursion. The domain of 𝐹 is a subclass of 𝐴. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}    &   𝐹 = 𝐵       dom 𝐹𝐴

Theoremfrrlem10 31035* Lemma for founded recursion. The union of all acceptable functions is a function. (Contributed by Paul Chapman, 21-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}    &   𝐹 = 𝐵       Fun 𝐹

Theoremfrrlem11 31036* Lemma for founded recursion. Here, we calculate the value of 𝐹 (the union of all acceptable functions). (Contributed by Paul Chapman, 21-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}    &   𝐹 = 𝐵       (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))

21.8.21  Surreal Numbers

Syntaxcsur 31037 Declare the class of all surreal numbers (see df-no 31040).
class No

Syntaxcslt 31038 Declare the less than relationship over surreal numbers (see df-slt 31041).
class <s

Syntaxcbday 31039 Declare the birthday function for surreal numbers (see df-bday 31042).
class bday

Definitiondf-no 31040* Define the class of surreal numbers. The surreal numbers are a proper class of numbers developed by John H. Conway and introduced by Donald Knuth in 1975. They form a proper class into which all ordered fields can be embedded. The approach we take to defining them was first introduced by Hary Goshnor, and is based on the conception of a "sign expansion" of a surreal number. We define the surreals as ordinal-indexed sequences of 1𝑜 and 2𝑜, analagous to Goshnor's ( − ) and ( + ).

After introducing this definition, we will abstract away from it using axioms that Norman Alling developed in "Foundations of Analysis over Surreal Number Fields." This is done in an effort to be agnostic towards the exact implementation of surreals. (Contributed by Scott Fenton, 9-Jun-2011.)

No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1𝑜, 2𝑜}}

Definitiondf-slt 31041* Next, we introduce surreal less-than, a comparison relationship over the surreals by lexicographically ordering them. (Contributed by Scott Fenton, 9-Jun-2011.)
<s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 No 𝑔 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)))}

Definitiondf-bday 31042 Finally, we introduce the birthday function. This function maps each surreal to an ordinal. In our implementation, this is the domain of the sign function. The important properties of this function are established later. (Contributed by Scott Fenton, 11-Jun-2011.)
bday = (𝑥 No ↦ dom 𝑥)

Theoremelno 31043* Membership in the surreals. (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
(𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})

Theoremsltval 31044* The value of the surreal less than relationship. (Contributed by Scott Fenton, 14-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))

Theorembdayval 31045 The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
(𝐴 No → ( bday 𝐴) = dom 𝐴)

Theoremnofun 31046 A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → Fun 𝐴)

Theoremnodmon 31047 The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → dom 𝐴 ∈ On)

Theoremnorn 31048 The range of a surreal is a subset of the surreal signs. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})

Theoremnofnbday 31049 A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No 𝐴 Fn ( bday 𝐴))

Theoremnodmord 31050 The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → Ord dom 𝐴)

Theoremelno2 31051 An alternative condition for membership in No . (Contributed by Scott Fenton, 21-Mar-2012.)
(𝐴 No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))

Theoremelno3 31052 Another condition for membership in No . (Contributed by Scott Fenton, 14-Apr-2012.)
(𝐴 No ↔ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ∧ dom 𝐴 ∈ On))

Theoremsltval2 31053* Alternate expression for surreal less than. Two surreals obey surreal less than iff they obey the sign ordering at the first place they differ. (Contributed by Scott Fenton, 17-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))

Theoremnofv 31054 The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.)
(𝐴 No → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))

Theoremnosgnn0 31055 is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ ∅ ∈ {1𝑜, 2𝑜}

Theoremnosgnn0i 31056 If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}       ∅ ≠ 𝑋

Theoremnoreson 31057 The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.)
((𝐴 No 𝐵 ∈ On) → (𝐴𝐵) ∈ No )

Theoremsltsgn1 31058* If 𝐴 <s 𝐵, then the sign of 𝐴 at the first place they differ is either undefined or 1𝑜. (Contributed by Scott Fenton, 4-Sep-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ((𝐴 {𝑘 ∈ On ∣ (𝐴𝑘) ≠ (𝐵𝑘)}) = ∅ ∨ (𝐴 {𝑘 ∈ On ∣ (𝐴𝑘) ≠ (𝐵𝑘)}) = 1𝑜)))

Theoremsltsgn2 31059* If 𝐴 <s 𝐵, then the sign of 𝐵 at the first place they differ is either undefined or 2𝑜. (Contributed by Scott Fenton, 4-Sep-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ((𝐵 {𝑘 ∈ On ∣ (𝐴𝑘) ≠ (𝐵𝑘)}) = ∅ ∨ (𝐵 {𝑘 ∈ On ∣ (𝐴𝑘) ≠ (𝐵𝑘)}) = 2𝑜)))

Theoremsltintdifex 31060* If 𝐴 <s 𝐵, then the intersection of all the ordinals that have differing signs in 𝐴 and 𝐵 exists. (Contributed by Scott Fenton, 22-Feb-2012.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V))

Theoremsltres 31061 If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)
((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → 𝐴 <s 𝐵))

Theoremnoxpsgn 31062 The Cartesian product of an ordinal and the singleton of a sign is a surreal. (Contributed by Scott Fenton, 21-Jun-2011.)
𝑋 ∈ {1𝑜, 2𝑜}       (𝐴 ∈ On → (𝐴 × {𝑋}) ∈ No )

Theoremnoxp1o 31063 The Cartesian product of an ordinal and {1𝑜} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
(𝐴 ∈ On → (𝐴 × {1𝑜}) ∈ No )

Theoremnoxp2o 31064 The Cartesian product of an ordinal and {2𝑜} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
(𝐴 ∈ On → (𝐴 × {2𝑜}) ∈ No )

Theoremnoseponlem 31065* Lemma for nosepon 31066. Consider a case of proper subset domain. (Contributed by Scott Fenton, 21-Sep-2020.)
((𝐴 No 𝐵 No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))

Theoremnosepon 31066* Given two unequal surreals, the minimal ordinal at which they differ is an ordinal. (Contributed by Scott Fenton, 21-Sep-2020.)
((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)

21.8.22  Surreal Numbers: Ordering

Theoremsltsolem1 31067 Lemma for sltso 31068. The sign expansion relationship totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.)
{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or ({1𝑜, 2𝑜} ∪ {∅})

Theoremsltso 31068 Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.)
<s Or No

Theoremsltirr 31069 Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → ¬ 𝐴 <s 𝐴)

Theoremslttr 31070 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 31071 Surreal less than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴))

Theoremslttri 31072 Surreal less than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵𝐴 = 𝐵𝐵 <s 𝐴))

Theoremslttrieq2 31073 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)))

21.8.23  Surreal Numbers: Birthday Function

Theorembdayfo 31074 The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
bday : No onto→On

Theorembdayfun 31075 The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
Fun bday

Theorembdayrn 31076 The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
ran bday = On

Theorembdaydm 31077 The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.)
dom bday = No

Theorembdayfn 31078 The birthday function is a function over No . (Contributed by Scott Fenton, 30-Jun-2011.)
bday Fn No

Theorembdayelon 31079 The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.)
( bday 𝐴) ∈ On

Theoremnoprc 31080 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ No ∈ V

21.8.24  Surreal Numbers: Density

Theoremfvnobday 31081 The value of a surreal at its birthday is . (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.)
(𝐴 No → (𝐴‘( bday 𝐴)) = ∅)

Theoremnodenselem3 31082* Lemma for nodense 31088. If one surreal is older than another, then there is an ordinal at which they are not equal. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ ( bday 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))

Theoremnodenselem4 31083* Lemma for nodense 31088. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)

Theoremnodenselem5 31084* Lemma for nodense 31088. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 31083 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
(((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))

Theoremnodenselem6 31085* The restriction of a surreal to the abstraction from nodenselem4 31083 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 ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ No )

Theoremnodenselem7 31086* Lemma for nodense 31088. 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 ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝐶) = (𝐵𝐶)))

Theoremnodenselem8 31087* Lemma for nodense 31088. 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 ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1𝑜 ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2𝑜)))

Theoremnodense 31088* Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD). (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 𝑥𝑥 <s 𝐵))

Theoremnocvxminlem 31089* Lemma for nocvxmin 31090. 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 31090* 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))

21.8.25  Surreal Numbers: Upper and Lower Bounds

Theoremnobndlem1 31091 Lemma for nobndup 31099 and nobnddown 31100. 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.)
(𝐴𝑉 → suc ( bday 𝐴) ∈ On)

Theoremnobndlem2 31092* Lemma for nobndup 31099 and nobnddown 31100. Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}       ((𝐹 No 𝐹𝐴) → 𝐶 ∈ On)

Theoremnobndlem3 31093* Lemma for nobndup 31099 and nobnddown 31100. Calculate the birthday of (𝐶 × {𝑋}). (Contributed by Scott Fenton, 17-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}       ((𝐹 No 𝐹𝐴) → ( bday ‘(𝐶 × {𝑋})) = 𝐶)

Theoremnobndlem4 31094* Lemma for nobndup 31099 and nobnddown 31100. The infimum of the class of all ordinals such that 𝐴 is not 𝑋 is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}       (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋} ∈ On)

Theoremnobndlem5 31095* Lemma for nobndup 31099 and nobnddown 31100. There is always a minimal value of a surreal that is not a given sign. (Contributed by Scott Fenton, 3-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}       (𝐴 No → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋}) ≠ 𝑋)

Theoremnobndlem6 31096* Lemma for nobndup 31099 and nobnddown 31100. Given an element 𝐴 of 𝐹, then the first position where it differs from 𝑋 is strictly less than 𝐶. (Contributed by Scott Fenton, 3-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}       ((𝐹 No 𝐴𝐹) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋} ∈ 𝐶)

Theoremnobndlem7 31097* Lemma for nobndup 31099 and nobnddown 31100. Calculate the value of (𝐶 × {𝑋}) at the minimal ordinal whose value is different from 𝑋. (Contributed by Scott Fenton, 3-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}       ((𝐹 No 𝐴𝐹) → ((𝐶 × {𝑋})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ 𝑋}) = 𝑋)

Theoremnobndlem8 31098* Lemma for nobndup 31099 and nobnddown 31100. Bound the birthday of (𝐶 × {𝑆}) above. (Contributed by Scott Fenton, 10-Apr-2017.)
𝑆 ∈ {1𝑜, 2𝑜}    &   𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑆}       ((𝐹 No 𝐹𝐴) → ( bday ‘(𝐶 × {𝑆})) ⊆ suc ( bday 𝐹))

Theoremnobndup 31099* Any set of surreals is bounded above by a surreal with a birthday no greater than the successor of their maximum birthday. (Contributed by Scott Fenton, 10-Apr-2017.)
((𝐴 No 𝐴𝑉) → ∃𝑥 No (∀𝑦𝐴 𝑦 <s 𝑥 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))

Theoremnobnddown 31100* Any set of surreals is bounded below by a surreal with a birthday no greater than the successor of their maximum birthday. (Contributed by Scott Fenton, 10-Apr-2017.)
((𝐴 No 𝐴𝑉) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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