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Theorem List for Metamath Proof Explorer - 42201-42300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
21.33.3  N-Digit Addition Proof Generator

This section formalizes theorems used in an n-digit addition proof generator.

Other theorems required: deccl 12566 addcomli 11281 00id 11264 addid1i 11276 addid2i 11277 eqid 2738 dec0h 12573 decadd 12605 decaddc 12606.

 
Theoremunitadd 42201 Theorem used in conjunction with decaddc 12606 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝐴 + 𝐡) = 𝐹    &   (𝐢 + 1) = 𝐡    &   π΄ ∈ β„•0    &   πΆ ∈ β„•0    β‡’   ((𝐴 + 𝐢) + 1) = 𝐹
 
21.33.4  AM-GM (for k = 2,3,4)
 
Theoremgsumws3 42202 Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    β‡’   ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐡 ∧ (𝑇 ∈ 𝐡 ∧ π‘ˆ ∈ 𝐡))) β†’ (𝐺 Ξ£g βŸ¨β€œπ‘†π‘‡π‘ˆβ€βŸ©) = (𝑆 + (𝑇 + π‘ˆ)))
 
Theoremgsumws4 42203 Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    β‡’   ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐡 ∧ (𝑇 ∈ 𝐡 ∧ (π‘ˆ ∈ 𝐡 ∧ 𝑉 ∈ 𝐡)))) β†’ (𝐺 Ξ£g βŸ¨β€œπ‘†π‘‡π‘ˆπ‘‰β€βŸ©) = (𝑆 + (𝑇 + (π‘ˆ + 𝑉))))
 
Theoremamgm2d 42204 Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 26261. (Contributed by Stanislas Polu, 8-Sep-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ+)    β‡’   (πœ‘ β†’ ((𝐴 Β· 𝐡)↑𝑐(1 / 2)) ≀ ((𝐴 + 𝐡) / 2))
 
Theoremamgm3d 42205 Arithmetic-geometric mean inequality for 𝑛 = 3. (Contributed by Stanislas Polu, 11-Sep-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ+)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    β‡’   (πœ‘ β†’ ((𝐴 Β· (𝐡 Β· 𝐢))↑𝑐(1 / 3)) ≀ ((𝐴 + (𝐡 + 𝐢)) / 3))
 
Theoremamgm4d 42206 Arithmetic-geometric mean inequality for 𝑛 = 4. (Contributed by Stanislas Polu, 11-Sep-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ+)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ 𝐷 ∈ ℝ+)    β‡’   (πœ‘ β†’ ((𝐴 Β· (𝐡 Β· (𝐢 Β· 𝐷)))↑𝑐(1 / 4)) ≀ ((𝐴 + (𝐡 + (𝐢 + 𝐷))) / 4))
 
21.34  Mathbox for Rohan Ridenour
 
21.34.1  Misc
 
TheoremspALT 42207 sp 2177 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2177 instead. (New usage is discouraged.)
(βˆ€π‘₯πœ‘ β†’ πœ‘)
 
Theoremelnelneqd 42208 Two classes are not equal if there is an element of one which is not an element of the other. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ Β¬ 𝐢 ∈ 𝐡)    β‡’   (πœ‘ β†’ Β¬ 𝐴 = 𝐡)
 
Theoremelnelneq2d 42209 Two classes are not equal if one but not the other is an element of a given class. (Contributed by Rohan Ridenour, 12-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ 𝐢)    &   (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐢)    β‡’   (πœ‘ β†’ Β¬ 𝐴 = 𝐡)
 
Theoremrr-spce 42210* Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.)
((πœ‘ ∧ π‘₯ = 𝐴) β†’ πœ“)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ βˆƒπ‘₯πœ“)
 
Theoremrexlimdvaacbv 42211* Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3152. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(π‘₯ = 𝑦 β†’ (πœ“ ↔ πœƒ))    &   ((πœ‘ ∧ (𝑦 ∈ 𝐴 ∧ πœƒ)) β†’ πœ’)    β‡’   (πœ‘ β†’ (βˆƒπ‘₯ ∈ 𝐴 πœ“ β†’ πœ’))
 
Theoremrexlimddvcbvw 42212* Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 42211. The equivalent of this theorem without the bound variable change is rexlimddv 3157. Version of rexlimddvcbv 42213 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by Gino Giotto, 2-Apr-2024.)
(πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 πœƒ)    &   ((πœ‘ ∧ (𝑦 ∈ 𝐴 ∧ πœ’)) β†’ πœ“)    &   (π‘₯ = 𝑦 β†’ (πœƒ ↔ πœ’))    β‡’   (πœ‘ β†’ πœ“)
 
Theoremrexlimddvcbv 42213* Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 42211. The equivalent of this theorem without the bound variable change is rexlimddv 3157. Usage of this theorem is discouraged because it depends on ax-13 2372, see rexlimddvcbvw 42212 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.)
(πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 πœƒ)    &   ((πœ‘ ∧ (𝑦 ∈ 𝐴 ∧ πœ’)) β†’ πœ“)    &   (π‘₯ = 𝑦 β†’ (πœƒ ↔ πœ’))    β‡’   (πœ‘ β†’ πœ“)
 
Theoremrr-elrnmpt3d 42214* Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   ((πœ‘ ∧ π‘₯ = 𝐢) β†’ 𝐡 = 𝐷)    β‡’   (πœ‘ β†’ 𝐷 ∈ ran 𝐹)
 
Theoremfinnzfsuppd 42215* If a function is zero outside of a finite set, it has finite support. (Contributed by Rohan Ridenour, 13-May-2024.)
(πœ‘ β†’ 𝐹 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹 Fn 𝐷)    &   (πœ‘ β†’ 𝑍 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐷) β†’ (π‘₯ ∈ 𝐴 ∨ (πΉβ€˜π‘₯) = 𝑍))    β‡’   (πœ‘ β†’ 𝐹 finSupp 𝑍)
 
Theoremrr-phpd 42216 Equivalent of php 9088 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ Ο‰)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    &   (πœ‘ β†’ 𝐴 β‰ˆ 𝐡)    β‡’   (πœ‘ β†’ 𝐴 = 𝐡)
 
Theoremsuceqd 42217 Deduction associated with suceq 6380. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ suc 𝐴 = suc 𝐡)
 
Theoremtfindsd 42218* Deduction associated with tfinds 7787. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(π‘₯ = βˆ… β†’ (πœ“ ↔ πœ’))    &   (π‘₯ = 𝑦 β†’ (πœ“ ↔ πœƒ))    &   (π‘₯ = suc 𝑦 β†’ (πœ“ ↔ 𝜏))    &   (π‘₯ = 𝐴 β†’ (πœ“ ↔ πœ‚))    &   (πœ‘ β†’ πœ’)    &   ((πœ‘ ∧ 𝑦 ∈ On ∧ πœƒ) β†’ 𝜏)    &   ((πœ‘ ∧ Lim π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ πœƒ) β†’ πœ“)    &   (πœ‘ β†’ 𝐴 ∈ On)    β‡’   (πœ‘ β†’ πœ‚)
 
21.34.2  Monoid rings
 
Syntaxcmnring 42219 Extend class notation with the monoid ring function.
class MndRing
 
Definitiondf-mnring 42220* Define the monoid ring function. This takes a monoid 𝑀 and a ring 𝑅 and produces a free left module over 𝑅 with a product extending the monoid function on 𝑀. (Contributed by Rohan Ridenour, 13-May-2024.)
MndRing = (π‘Ÿ ∈ V, π‘š ∈ V ↦ ⦋(π‘Ÿ freeLMod (Baseβ€˜π‘š)) / π‘£β¦Œ(𝑣 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ (Baseβ€˜π‘£), 𝑦 ∈ (Baseβ€˜π‘£) ↦ (𝑣 Ξ£g (π‘Ž ∈ (Baseβ€˜π‘š), 𝑏 ∈ (Baseβ€˜π‘š) ↦ (𝑖 ∈ (Baseβ€˜π‘š) ↦ if(𝑖 = (π‘Ž(+gβ€˜π‘š)𝑏), ((π‘₯β€˜π‘Ž)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘)), (0gβ€˜π‘Ÿ))))))⟩))
 
Theoremmnringvald 42221* Value of the monoid ring function. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π΄ = (Baseβ€˜π‘€)    &    + = (+gβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐴)    &   π΅ = (Baseβ€˜π‘‰)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝐹 = (𝑉 sSet ⟨(.rβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝑉 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 )))))⟩))
 
Theoremmnringnmulrd 42222 Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (Revised by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   πΈ = Slot (πΈβ€˜ndx)    &   (πΈβ€˜ndx) β‰  (.rβ€˜ndx)    &   π΄ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐴)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ (πΈβ€˜π‘‰) = (πΈβ€˜πΉ))
 
TheoremmnringnmulrdOLD 42223 Obsolete version of mnringnmulrd 42222 as of 1-Nov-2024. Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑅 MndRing 𝑀)    &   πΈ = Slot 𝑁    &   π‘ ∈ β„•    &   π‘ β‰  (.rβ€˜ndx)    &   π΄ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐴)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ (πΈβ€˜π‘‰) = (πΈβ€˜πΉ))
 
Theoremmnringbased 42224 The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΄ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐴)    &   π΅ = (Baseβ€˜π‘‰)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΉ))
 
TheoremmnringbasedOLD 42225 Obsolete version of mnringnmulrd 42222 as of 1-Nov-2024. The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΄ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐴)    &   π΅ = (Baseβ€˜π‘‰)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΉ))
 
Theoremmnringbaserd 42226 The base set of a monoid ring. Converse of mnringbased 42224. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΅ = (Baseβ€˜πΉ)    &   π΄ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐴)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘‰))
 
Theoremmnringelbased 42227 Membership in the base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΅ = (Baseβ€˜πΉ)    &   π΄ = (Baseβ€˜π‘€)    &   πΆ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ (𝑋 ∈ 𝐡 ↔ (𝑋 ∈ (𝐢 ↑m 𝐴) ∧ 𝑋 finSupp 0 )))
 
Theoremmnringbasefd 42228 Elements of a monoid ring are functions. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΅ = (Baseβ€˜πΉ)    &   π΄ = (Baseβ€˜π‘€)    &   πΆ = (Baseβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ 𝑋:𝐴⟢𝐢)
 
Theoremmnringbasefsuppd 42229 Elements of a monoid ring are finitely supported. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΅ = (Baseβ€˜πΉ)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ 𝑋 finSupp 0 )
 
Theoremmnringaddgd 42230 The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΄ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐴)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ (+gβ€˜π‘‰) = (+gβ€˜πΉ))
 
TheoremmnringaddgdOLD 42231 Obsolete version of mnringaddgd 42230 as of 1-Nov-2024. The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΄ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐴)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ (+gβ€˜π‘‰) = (+gβ€˜πΉ))
 
Theoremmnring0gd 42232 The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΄ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐴)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ (0gβ€˜π‘‰) = (0gβ€˜πΉ))
 
Theoremmnring0g2d 42233 The additive identity of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &    0 = (0gβ€˜π‘…)    &   π΄ = (Baseβ€˜π‘€)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ (𝐴 Γ— { 0 }) = (0gβ€˜πΉ))
 
Theoremmnringmulrd 42234* The ring product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΅ = (Baseβ€˜πΉ)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π΄ = (Baseβ€˜π‘€)    &    + = (+gβ€˜π‘€)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (𝐹 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘₯β€˜π‘Ž) Β· (π‘¦β€˜π‘)), 0 ))))) = (.rβ€˜πΉ))
 
Theoremmnringscad 42235 The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝑅 = (Scalarβ€˜πΉ))
 
TheoremmnringscadOLD 42236 Obsolete version of mnringscad 42235 as of 1-Nov-2024. The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑅 MndRing 𝑀)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝑅 = (Scalarβ€˜πΉ))
 
Theoremmnringvscad 42237 The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΅ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐡)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ ( ·𝑠 β€˜π‘‰) = ( ·𝑠 β€˜πΉ))
 
TheoremmnringvscadOLD 42238 Obsolete version of mnringvscad 42237 as of 1-Nov-2024. The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΅ = (Baseβ€˜π‘€)    &   π‘‰ = (𝑅 freeLMod 𝐡)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    β‡’   (πœ‘ β†’ ( ·𝑠 β€˜π‘‰) = ( ·𝑠 β€˜πΉ))
 
Theoremmnringlmodd 42239 Monoid rings are left modules. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑀 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ 𝐹 ∈ LMod)
 
Theoremmnringmulrvald 42240* Value of multiplication in a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΅ = (Baseβ€˜πΉ)    &    βˆ™ = (.rβ€˜π‘…)    &    𝟎 = (0gβ€˜π‘…)    &   π΄ = (Baseβ€˜π‘€)    &    + = (+gβ€˜π‘€)    &    Β· = (.rβ€˜πΉ)    &   (πœ‘ β†’ 𝑅 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑀 ∈ π‘Š)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· π‘Œ) = (𝐹 Ξ£g (π‘Ž ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (π‘Ž + 𝑏), ((π‘‹β€˜π‘Ž) βˆ™ (π‘Œβ€˜π‘)), 𝟎 )))))
 
Theoremmnringmulrcld 42241 Monoid rings are closed under multiplication. (Contributed by Rohan Ridenour, 14-May-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   π΅ = (Baseβ€˜πΉ)    &   π΄ = (Baseβ€˜π‘€)    &    Β· = (.rβ€˜πΉ)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑀 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· π‘Œ) ∈ 𝐡)
 
21.34.3  Shorter primitive equivalent of ax-groth
 
21.34.3.1  Grothendieck universes are closed under collection
 
Theoremgru0eld 42242 A nonempty Grothendieck universe contains the empty set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ 𝐺 ∈ Univ)    &   (πœ‘ β†’ 𝐴 ∈ 𝐺)    β‡’   (πœ‘ β†’ βˆ… ∈ 𝐺)
 
Theoremgrusucd 42243 Grothendieck universes are closed under ordinal successor. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(πœ‘ β†’ 𝐺 ∈ Univ)    &   (πœ‘ β†’ 𝐴 ∈ 𝐺)    β‡’   (πœ‘ β†’ suc 𝐴 ∈ 𝐺)
 
Theoremr1rankcld 42244 Any rank of the cumulative hierarchy is closed under the rank function. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ (𝑅1β€˜π‘…))    β‡’   (πœ‘ β†’ (rankβ€˜π΄) ∈ (𝑅1β€˜π‘…))
 
Theoremgrur1cld 42245 Grothendieck universes are closed under the cumulative hierarchy function. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(πœ‘ β†’ 𝐺 ∈ Univ)    &   (πœ‘ β†’ 𝐴 ∈ 𝐺)    β‡’   (πœ‘ β†’ (𝑅1β€˜π΄) ∈ 𝐺)
 
Theoremgrurankcld 42246 Grothendieck universes are closed under the rank function. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(πœ‘ β†’ 𝐺 ∈ Univ)    &   (πœ‘ β†’ 𝐴 ∈ 𝐺)    β‡’   (πœ‘ β†’ (rankβ€˜π΄) ∈ 𝐺)
 
Theoremgrurankrcld 42247 If a Grothendieck universe contains a set's rank, it contains that set. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(πœ‘ β†’ 𝐺 ∈ Univ)    &   (πœ‘ β†’ (rankβ€˜π΄) ∈ 𝐺)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝐴 ∈ 𝐺)
 
Syntaxcscott 42248 Extend class notation with the Scott's trick operation.
class Scott 𝐴
 
Definitiondf-scott 42249* Define the Scott operation. This operation constructs a subset of the input class which is nonempty whenever its input is using Scott's trick. (Contributed by Rohan Ridenour, 9-Aug-2023.)
Scott 𝐴 = {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)}
 
Theoremscotteqd 42250 Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ Scott 𝐴 = Scott 𝐡)
 
Theoremscotteq 42251 Closed form of scotteqd 42250. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝐴 = 𝐡 β†’ Scott 𝐴 = Scott 𝐡)
 
Theoremnfscott 42252 Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
β„²π‘₯𝐴    β‡’   β„²π‘₯Scott 𝐴
 
Theoremscottabf 42253* Value of the Scott operation at a class abstraction. Variant of scottab 42254 with a nonfreeness hypothesis instead of a disjoint variable condition. (Contributed by Rohan Ridenour, 14-Aug-2023.)
β„²π‘₯πœ“    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   Scott {π‘₯ ∣ πœ‘} = {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
 
Theoremscottab 42254* Value of the Scott operation at a class abstraction. (Contributed by Rohan Ridenour, 14-Aug-2023.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   Scott {π‘₯ ∣ πœ‘} = {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
 
Theoremscottabes 42255* Value of the Scott operation at a class abstraction. Variant of scottab 42254 using explicit substitution. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Scott {π‘₯ ∣ πœ‘} = {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦([𝑦 / π‘₯]πœ‘ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
 
Theoremscottss 42256 Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Scott 𝐴 βŠ† 𝐴
 
Theoremelscottab 42257* An element of the output of the Scott operation applied to a class abstraction satisfies the class abstraction's predicate. (Contributed by Rohan Ridenour, 14-Aug-2023.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (𝑦 ∈ Scott {π‘₯ ∣ πœ‘} β†’ πœ“)
 
Theoremscottex2 42258 scottex 9755 expressed using Scott. (Contributed by Rohan Ridenour, 9-Aug-2023.)
Scott 𝐴 ∈ V
 
Theoremscotteld 42259* The Scott operation sends inhabited classes to inhabited sets. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ Scott 𝐴)
 
Theoremscottelrankd 42260 Property of a Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ 𝐡 ∈ Scott 𝐴)    &   (πœ‘ β†’ 𝐢 ∈ Scott 𝐴)    β‡’   (πœ‘ β†’ (rankβ€˜π΅) βŠ† (rankβ€˜πΆ))
 
Theoremscottrankd 42261 Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ 𝐡 ∈ Scott 𝐴)    β‡’   (πœ‘ β†’ (rankβ€˜Scott 𝐴) = suc (rankβ€˜π΅))
 
Theoremgruscottcld 42262 If a Grothendieck universe contains an element of a Scott's trick set, it contains the Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ 𝐺 ∈ Univ)    &   (πœ‘ β†’ 𝐡 ∈ 𝐺)    &   (πœ‘ β†’ 𝐡 ∈ Scott 𝐴)    β‡’   (πœ‘ β†’ Scott 𝐴 ∈ 𝐺)
 
Syntaxccoll 42263 Extend class notation with the collection operation.
class (𝐹 Coll 𝐴)
 
Definitiondf-coll 42264* Define the collection operation. This is similar to the image set operation β€œ, but it uses Scott's trick to ensure the output is always a set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 Coll 𝐴) = βˆͺ π‘₯ ∈ 𝐴 Scott (𝐹 β€œ {π‘₯})
 
Theoremdfcoll2 42265* Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 Coll 𝐴) = βˆͺ π‘₯ ∈ 𝐴 Scott {𝑦 ∣ π‘₯𝐹𝑦}
 
Theoremcolleq12d 42266 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ 𝐹 = 𝐺)    &   (πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ (𝐹 Coll 𝐴) = (𝐺 Coll 𝐡))
 
Theoremcolleq1 42267 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 = 𝐺 β†’ (𝐹 Coll 𝐴) = (𝐺 Coll 𝐴))
 
Theoremcolleq2 42268 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐴 = 𝐡 β†’ (𝐹 Coll 𝐴) = (𝐹 Coll 𝐡))
 
Theoremnfcoll 42269 Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
β„²π‘₯𝐹    &   β„²π‘₯𝐴    β‡’   β„²π‘₯(𝐹 Coll 𝐴)
 
Theoremcollexd 42270 The output of the collection operation is a set if the second input is. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐹 Coll 𝐴) ∈ V)
 
Theoremcpcolld 42271* Property of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ π‘₯𝐹𝑦)    β‡’   (πœ‘ β†’ βˆƒπ‘¦ ∈ (𝐹 Coll 𝐴)π‘₯𝐹𝑦)
 
Theoremcpcoll2d 42272* cpcolld 42271 with an extra existential quantifier. (Contributed by Rohan Ridenour, 12-Aug-2023.)
(πœ‘ β†’ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘¦ π‘₯𝐹𝑦)    β‡’   (πœ‘ β†’ βˆƒπ‘¦ ∈ (𝐹 Coll 𝐴)π‘₯𝐹𝑦)
 
Theoremgrucollcld 42273 A Grothendieck universe contains the output of a collection operation whenever its left input is a relation on the universe, and its right input is in the universe. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(πœ‘ β†’ 𝐺 ∈ Univ)    &   (πœ‘ β†’ 𝐹 βŠ† (𝐺 Γ— 𝐺))    &   (πœ‘ β†’ 𝐴 ∈ 𝐺)    β‡’   (πœ‘ β†’ (𝐹 Coll 𝐴) ∈ 𝐺)
 
21.34.3.2  Minimal universes
 
Theoremismnu 42274* The hypothesis of this theorem defines a class M of sets that we temporarily call "minimal universes", and which will turn out in grumnueq 42300 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 42312, except for the π‘₯ ∈ 𝑦 clause.

A minimal universe is closed under subsets (mnussd 42276), powersets (mnupwd 42280), and an operation which is similar to a combination of collection and union (mnuop3d 42284), from which closure under pairing (mnuprd 42289), unions (mnuunid 42290), and function ranges (mnurnd 42296) can be deduced, from which equivalence with Grothendieck universes (grumnueq 42300) can be deduced. (Contributed by Rohan Ridenour, 13-Aug-2023.)

𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    β‡’   (π‘ˆ ∈ 𝑉 β†’ (π‘ˆ ∈ 𝑀 ↔ βˆ€π‘§ ∈ π‘ˆ (𝒫 𝑧 βŠ† π‘ˆ ∧ βˆ€π‘“βˆƒπ‘€ ∈ π‘ˆ (𝒫 𝑧 βŠ† 𝑀 ∧ βˆ€π‘– ∈ 𝑧 (βˆƒπ‘£ ∈ π‘ˆ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) β†’ βˆƒπ‘’ ∈ 𝑓 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀))))))
 
Theoremmnuop123d 42275* Operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (𝒫 𝐴 βŠ† π‘ˆ ∧ βˆ€π‘“βˆƒπ‘€ ∈ π‘ˆ (𝒫 𝐴 βŠ† 𝑀 ∧ βˆ€π‘– ∈ 𝐴 (βˆƒπ‘£ ∈ π‘ˆ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) β†’ βˆƒπ‘’ ∈ 𝑓 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀)))))
 
Theoremmnussd 42276* Minimal universes are closed under subsets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    β‡’   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)
 
Theoremmnuss2d 42277* mnussd 42276 with arguments provided with an existential quantifier. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ π‘ˆ 𝐴 βŠ† π‘₯)    β‡’   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)
 
Theoremmnu0eld 42278* A nonempty minimal universe contains the empty set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ βˆ… ∈ π‘ˆ)
 
Theoremmnuop23d 42279* Second and third operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐹 ∈ 𝑉)    β‡’   (πœ‘ β†’ βˆƒπ‘€ ∈ π‘ˆ (𝒫 𝐴 βŠ† 𝑀 ∧ βˆ€π‘– ∈ 𝐴 (βˆƒπ‘£ ∈ π‘ˆ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹) β†’ βˆƒπ‘’ ∈ 𝐹 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀))))
 
Theoremmnupwd 42280* Minimal universes are closed under powersets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ 𝒫 𝐴 ∈ π‘ˆ)
 
Theoremmnusnd 42281* Minimal universes are closed under singletons. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ {𝐴} ∈ π‘ˆ)
 
Theoremmnuprssd 42282* A minimal universe contains pairs of subsets of an element of the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐢 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐢)    β‡’   (πœ‘ β†’ {𝐴, 𝐡} ∈ π‘ˆ)
 
Theoremmnuprss2d 42283* Special case of mnuprssd 42282. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐢 ∈ π‘ˆ)    &   π΄ βŠ† 𝐢    &   π΅ βŠ† 𝐢    β‡’   (πœ‘ β†’ {𝐴, 𝐡} ∈ π‘ˆ)
 
Theoremmnuop3d 42284* Third operation of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐹 βŠ† π‘ˆ)    β‡’   (πœ‘ β†’ βˆƒπ‘€ ∈ π‘ˆ βˆ€π‘– ∈ 𝐴 (βˆƒπ‘£ ∈ 𝐹 𝑖 ∈ 𝑣 β†’ βˆƒπ‘’ ∈ 𝐹 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀)))
 
Theoremmnuprdlem1 42285* Lemma for mnuprd 42289. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{βˆ…, {𝐴}}, {{βˆ…}, {𝐡}}}    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘– ∈ {βˆ…, {βˆ…}}βˆƒπ‘’ ∈ 𝐹 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀))    β‡’   (πœ‘ β†’ 𝐴 ∈ 𝑀)
 
Theoremmnuprdlem2 42286* Lemma for mnuprd 42289. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{βˆ…, {𝐴}}, {{βˆ…}, {𝐡}}}    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    &   (πœ‘ β†’ Β¬ 𝐴 = βˆ…)    &   (πœ‘ β†’ βˆ€π‘– ∈ {βˆ…, {βˆ…}}βˆƒπ‘’ ∈ 𝐹 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀))    β‡’   (πœ‘ β†’ 𝐡 ∈ 𝑀)
 
Theoremmnuprdlem3 42287* Lemma for mnuprd 42289. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{βˆ…, {𝐴}}, {{βˆ…}, {𝐡}}}    &   β„²π‘–πœ‘    β‡’   (πœ‘ β†’ βˆ€π‘– ∈ {βˆ…, {βˆ…}}βˆƒπ‘£ ∈ 𝐹 𝑖 ∈ 𝑣)
 
Theoremmnuprdlem4 42288* Lemma for mnuprd 42289. General case. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   πΉ = {{βˆ…, {𝐴}}, {{βˆ…}, {𝐡}}}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    &   (πœ‘ β†’ Β¬ 𝐴 = βˆ…)    β‡’   (πœ‘ β†’ {𝐴, 𝐡} ∈ π‘ˆ)
 
Theoremmnuprd 42289* Minimal universes are closed under pairing. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ {𝐴, 𝐡} ∈ π‘ˆ)
 
Theoremmnuunid 42290* Minimal universes are closed under union. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ βˆͺ 𝐴 ∈ π‘ˆ)
 
Theoremmnuund 42291* Minimal universes are closed under binary unions. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (𝐴 βˆͺ 𝐡) ∈ π‘ˆ)
 
Theoremmnutrcld 42292* Minimal universes contain the elements of their elements. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    β‡’   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)
 
Theoremmnutrd 42293* Minimal universes are transitive. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    β‡’   (πœ‘ β†’ Tr π‘ˆ)
 
Theoremmnurndlem1 42294* Lemma for mnurnd 42296. (Contributed by Rohan Ridenour, 12-Aug-2023.)
(πœ‘ β†’ 𝐹:π΄βŸΆπ‘ˆ)    &   π΄ ∈ V    &   (πœ‘ β†’ βˆ€π‘– ∈ 𝐴 (βˆƒπ‘£ ∈ ran (π‘Ž ∈ 𝐴 ↦ {π‘Ž, {(πΉβ€˜π‘Ž), 𝐴}})𝑖 ∈ 𝑣 β†’ βˆƒπ‘’ ∈ ran (π‘Ž ∈ 𝐴 ↦ {π‘Ž, {(πΉβ€˜π‘Ž), 𝐴}})(𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀)))    β‡’   (πœ‘ β†’ ran 𝐹 βŠ† 𝑀)
 
Theoremmnurndlem2 42295* Lemma for mnurnd 42296. Deduction theorem input. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐹:π΄βŸΆπ‘ˆ)    &   π΄ ∈ V    β‡’   (πœ‘ β†’ ran 𝐹 ∈ π‘ˆ)
 
Theoremmnurnd 42296* Minimal universes contain ranges of functions from an element of the universe to the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐹:π΄βŸΆπ‘ˆ)    β‡’   (πœ‘ β†’ ran 𝐹 ∈ π‘ˆ)
 
Theoremmnugrud 42297* Minimal universes are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ π‘ˆ ∈ 𝑀)    β‡’   (πœ‘ β†’ π‘ˆ ∈ Univ)
 
Theoremgrumnudlem 42298* Lemma for grumnud 42299. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ 𝐺 ∈ Univ)    &   πΉ = ({βŸ¨π‘, π‘βŸ© ∣ βˆƒπ‘‘(βˆͺ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 Γ— 𝐺))    &   ((𝑖 ∈ 𝐺 ∧ β„Ž ∈ 𝐺) β†’ (π‘–πΉβ„Ž ↔ βˆƒπ‘—(βˆͺ 𝑗 = β„Ž ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))    &   ((β„Ž ∈ (𝐹 Coll 𝑧) ∧ (βˆͺ 𝑗 = β„Ž ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) β†’ βˆƒπ‘’ ∈ 𝑓 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 ∈ (𝐹 Coll 𝑧)))    β‡’   (πœ‘ β†’ 𝐺 ∈ 𝑀)
 
Theoremgrumnud 42299* Grothendieck universes are minimal universes. (Contributed by Rohan Ridenour, 12-Aug-2023.)
𝑀 = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}    &   (πœ‘ β†’ 𝐺 ∈ Univ)    β‡’   (πœ‘ β†’ 𝐺 ∈ 𝑀)
 
Theoremgrumnueq 42300* The class of Grothendieck universes is equal to the class of minimal universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Univ = {π‘˜ ∣ βˆ€π‘™ ∈ π‘˜ (𝒫 𝑙 βŠ† π‘˜ ∧ βˆ€π‘šβˆƒπ‘› ∈ π‘˜ (𝒫 𝑙 βŠ† 𝑛 ∧ βˆ€π‘ ∈ 𝑙 (βˆƒπ‘ž ∈ π‘˜ (𝑝 ∈ π‘ž ∧ π‘ž ∈ π‘š) β†’ βˆƒπ‘Ÿ ∈ π‘š (𝑝 ∈ π‘Ÿ ∧ βˆͺ π‘Ÿ βŠ† 𝑛))))}
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 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 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46966
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