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Theorem List for Metamath Proof Explorer - 40401-40500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsuprlubrd 40401* Natural deduction form of specialized suprlub 11594. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑧𝐴 𝐵 < 𝑧)       (𝜑𝐵 < sup(𝐴, ℝ, < ))
 
Theoremimo72b2lem1 40402* Lemma for imo72b2 40406. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑 → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)    &   (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)       (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
 
Theoremlemuldiv3d 40403 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑 → (𝐵 · 𝐴) ≤ 𝐶)    &   (𝜑 → 0 < 𝐴)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑𝐵 ≤ (𝐶 / 𝐴))
 
Theoremlemuldiv4d 40404 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐵 ≤ (𝐶 / 𝐴))    &   (𝜑 → 0 < 𝐴)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵 · 𝐴) ≤ 𝐶)
 
Theoremrspcdvinvd 40405* If something is true for all then it's true for some class. (Contributed by Stanislas Polu, 9-Mar-2020.)
((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝐴𝐵)    &   (𝜑 → ∀𝑥𝐵 𝜓)       (𝜑𝜒)
 
Theoremimo72b2 40406* IMO 1972 B2. (14th International Mathemahics Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐺:ℝ⟶ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))    &   (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)    &   (𝜑 → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)       (𝜑 → (abs‘(𝐺𝐵)) ≤ 1)
 
20.31.2  INT Inequalities Proof Generator

This section formalizes theorems necessary to reproduce the equality and inequality generator described in "Neural Theorem Proving on Inequality Problems" http://aitp-conference.org/2020/abstract/paper_18.pdf.

Other theorems required: 0red 10633 1red 10631 readdcld 10659 remulcld 10660 eqcomd 2827.

 
Theoremint-addcomd 40407 AdditionCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐴))
 
Theoremint-addassocd 40408 AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))
 
Theoremint-addsimpd 40409 AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → 0 = (𝐴𝐵))
 
Theoremint-mulcomd 40410 MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴))
 
Theoremint-mulassocd 40411 MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))
 
Theoremint-mulsimpd 40412 MultiplicationSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐵 ≠ 0)       (𝜑 → 1 = (𝐴 / 𝐵))
 
Theoremint-leftdistd 40413 AdditionMultiplicationLeftDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ((𝐶 + 𝐷) · 𝐵) = ((𝐶 · 𝐴) + (𝐷 · 𝐴)))
 
Theoremint-rightdistd 40414 AdditionMultiplicationRightDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 · (𝐶 + 𝐷)) = ((𝐴 · 𝐶) + (𝐴 · 𝐷)))
 
Theoremint-sqdefd 40415 SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 · 𝐵) = (𝐴↑2))
 
Theoremint-mul11d 40416 First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 · 1) = 𝐵)
 
Theoremint-mul12d 40417 Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (1 · 𝐴) = 𝐵)
 
Theoremint-add01d 40418 First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 + 0) = 𝐵)
 
Theoremint-add02d 40419 Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (0 + 𝐴) = 𝐵)
 
Theoremint-sqgeq0d 40420 SquareGEQZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → 0 ≤ (𝐴 · 𝐵))
 
Theoremint-eqprincd 40421 PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))
 
Theoremint-eqtransd 40422 EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴 = 𝐶)
 
Theoremint-eqmvtd 40423 EquMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐴 = (𝐶 + 𝐷))       (𝜑𝐶 = (𝐵𝐷))
 
Theoremint-eqineqd 40424 EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑𝐵𝐴)
 
Theoremint-ineqmvtd 40425 IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑𝐴 = (𝐶 + 𝐷))       (𝜑 → (𝐵𝐷) ≤ 𝐶)
 
Theoremint-ineq1stprincd 40426 FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑𝐷𝐶)       (𝜑 → (𝐵 + 𝐷) ≤ (𝐴 + 𝐶))
 
Theoremint-ineq2ndprincd 40427 SecondPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑 → 0 ≤ 𝐶)       (𝜑 → (𝐵 · 𝐶) ≤ (𝐴 · 𝐶))
 
Theoremint-ineqtransd 40428 InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐵)       (𝜑𝐶𝐴)
 
20.31.3  N-Digit Addition Proof Generator

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

Other theorems required: deccl 12102 addcomli 10821 00id 10804 addid1i 10816 addid2i 10817 eqid 2821 dec0h 12109 decadd 12141 decaddc 12142.

 
Theoremunitadd 40429 Theorem used in conjunction with decaddc 12142 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝐴 + 𝐵) = 𝐹    &   (𝐶 + 1) = 𝐵    &   𝐴 ∈ ℕ0    &   𝐶 ∈ ℕ0       ((𝐴 + 𝐶) + 1) = 𝐹
 
20.31.4  AM-GM (for k = 2,3,4)
 
Theoremgsumws3 40430 Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑆𝐵 ∧ (𝑇𝐵𝑈𝐵))) → (𝐺 Σg ⟨“𝑆𝑇𝑈”⟩) = (𝑆 + (𝑇 + 𝑈)))
 
Theoremgsumws4 40431 Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑆𝐵 ∧ (𝑇𝐵 ∧ (𝑈𝐵𝑉𝐵)))) → (𝐺 Σg ⟨“𝑆𝑇𝑈𝑉”⟩) = (𝑆 + (𝑇 + (𝑈 + 𝑉))))
 
Theoremamgm2d 40432 Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 25495. (Contributed by Stanislas Polu, 8-Sep-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2))
 
Theoremamgm3d 40433 Arithmetic-geometric mean inequality for 𝑛 = 3. (Contributed by Stanislas Polu, 11-Sep-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3)) ≤ ((𝐴 + (𝐵 + 𝐶)) / 3))
 
Theoremamgm4d 40434 Arithmetic-geometric mean inequality for 𝑛 = 4. (Contributed by Stanislas Polu, 11-Sep-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ+)       (𝜑 → ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4)) ≤ ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4))
 
20.32  Mathbox for Rohan Ridenour
 
20.32.1  Misc
 
TheoremspALT 40435 sp 2172 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2172 instead. (New usage is discouraged.)
(∀𝑥𝜑𝜑)
 
Theoremelnelneqd 40436 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.)
(𝜑𝐶𝐴)    &   (𝜑 → ¬ 𝐶𝐵)       (𝜑 → ¬ 𝐴 = 𝐵)
 
Theoremelnelneq2d 40437 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.)
(𝜑𝐴𝐶)    &   (𝜑 → ¬ 𝐵𝐶)       (𝜑 → ¬ 𝐴 = 𝐵)
 
Theoremrr-spce 40438* Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.)
((𝜑𝑥 = 𝐴) → 𝜓)    &   (𝜑𝐴𝑉)       (𝜑 → ∃𝑥𝜓)
 
Theoremrexlimdvaacbv 40439* Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3285. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝑥 = 𝑦 → (𝜓𝜃))    &   ((𝜑 ∧ (𝑦𝐴𝜃)) → 𝜒)       (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 
Theoremrexlimddvcbv 40440* Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40439. The equivalent of this theorem without the bound variable change is rexlimddv 3291. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑 → ∃𝑥𝐴 𝜃)    &   ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜃𝜒))       (𝜑𝜓)
 
Theoremrr-elrnmpt3d 40441* Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷𝑉)    &   ((𝜑𝑥 = 𝐶) → 𝐵 = 𝐷)       (𝜑𝐷 ∈ ran 𝐹)
 
Theoremrr-phpd 40442 Equivalent of php 8690 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐴 ∈ ω)    &   (𝜑𝐵𝐴)    &   (𝜑𝐴𝐵)       (𝜑𝐴 = 𝐵)
 
Theoremsuceqd 40443 Deduction associated with suceq 6250. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(𝜑𝐴 = 𝐵)       (𝜑 → suc 𝐴 = suc 𝐵)
 
Theoremtfindsd 40444* Deduction associated with tfinds 7562. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(𝑥 = ∅ → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = suc 𝑦 → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   ((𝜑𝑦 ∈ On ∧ 𝜃) → 𝜏)    &   ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦𝑥 𝜃) → 𝜓)    &   (𝜑𝐴 ∈ On)       (𝜑𝜂)
 
20.32.2  Shorter primitive equivalent of ax-groth
 
20.32.2.1  Grothendieck universes are closed under collection
 
Theoremgru0eld 40445 A nonempty Grothendieck universe contains the empty set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → ∅ ∈ 𝐺)
 
Theoremgrusucd 40446 Grothendieck universes are closed under ordinal successor. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → suc 𝐴𝐺)
 
Theoremr1rankcld 40447 Any rank of the cumulative hierarchy is closed under the rank function. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐴 ∈ (𝑅1𝑅))       (𝜑 → (rank‘𝐴) ∈ (𝑅1𝑅))
 
Theoremgrur1cld 40448 Grothendieck universes are closed under the cumulative hierarchy function. (Contributed by Rohan Ridenour, 8-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → (𝑅1𝐴) ∈ 𝐺)
 
Theoremgrurankcld 40449 Grothendieck universes are closed under the rank function. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐴𝐺)       (𝜑 → (rank‘𝐴) ∈ 𝐺)
 
Theoremgrurankrcld 40450 If a Grothendieck universe contains a set's rank, it contains that set. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐺 ∈ Univ)    &   (𝜑 → (rank‘𝐴) ∈ 𝐺)    &   (𝜑𝐴𝑉)       (𝜑𝐴𝐺)
 
Syntaxcscott 40451 Extend class notation with the Scott's trick operation.
class Scott 𝐴
 
Definitiondf-scott 40452* 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 = {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
 
Theoremscotteqd 40453 Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝜑𝐴 = 𝐵)       (𝜑 → Scott 𝐴 = Scott 𝐵)
 
Theoremscotteq 40454 Closed form of scotteqd 40453. (Contributed by Rohan Ridenour, 9-Aug-2023.)
(𝐴 = 𝐵 → Scott 𝐴 = Scott 𝐵)
 
Theoremnfscott 40455 Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝑥𝐴       𝑥Scott 𝐴
 
Theoremscottabf 40456* Value of the Scott operation at a class abstraction. Variant of scottab 40457 with a nonfreeness hypothesis instead of a disjoint variable condition. (Contributed by Rohan Ridenour, 14-Aug-2023.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
 
Theoremscottab 40457* Value of the Scott operation at a class abstraction. (Contributed by Rohan Ridenour, 14-Aug-2023.)
(𝑥 = 𝑦 → (𝜑𝜓))       Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
 
Theoremscottabes 40458* Value of the Scott operation at a class abstraction. Variant of scottab 40457 using explicit substitution. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
 
Theoremscottss 40459 Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Scott 𝐴𝐴
 
Theoremelscottab 40460* 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.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑦 ∈ Scott {𝑥𝜑} → 𝜓)
 
Theoremscottex2 40461 scottex 9303 expressed using Scott. (Contributed by Rohan Ridenour, 9-Aug-2023.)
Scott 𝐴 ∈ V
 
Theoremscotteld 40462* The Scott operation sends inhabited classes to inhabited sets. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑 → ∃𝑥 𝑥𝐴)       (𝜑 → ∃𝑥 𝑥 ∈ Scott 𝐴)
 
Theoremscottelrankd 40463 Property of a Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐵 ∈ Scott 𝐴)    &   (𝜑𝐶 ∈ Scott 𝐴)       (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶))
 
Theoremscottrankd 40464 Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐵 ∈ Scott 𝐴)       (𝜑 → (rank‘Scott 𝐴) = suc (rank‘𝐵))
 
Theoremgruscottcld 40465 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.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐵𝐺)    &   (𝜑𝐵 ∈ Scott 𝐴)       (𝜑 → Scott 𝐴𝐺)
 
Syntaxccoll 40466 Extend class notation with the collection operation.
class (𝐹 Coll 𝐴)
 
Definitiondf-coll 40467* Define the collection operation. This is similar to the image set operation , but it uses Scott's trick to ensure the output is always a set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 Coll 𝐴) = 𝑥𝐴 Scott (𝐹 “ {𝑥})
 
Theoremdfcoll2 40468* Alternate definition of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑦𝑥𝐹𝑦}
 
Theoremcolleq12d 40469 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))
 
Theoremcolleq1 40470 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐹 = 𝐺 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐴))
 
Theoremcolleq2 40471 Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵))
 
Theoremnfcoll 40472 Bound-variable hypothesis builder for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹 Coll 𝐴)
 
Theoremcollexd 40473 The output of the collection operation is a set if the second input is. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝐴𝑉)       (𝜑 → (𝐹 Coll 𝐴) ∈ V)
 
Theoremcpcolld 40474* Property of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐹𝑦)       (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
 
Theoremcpcoll2d 40475* cpcolld 40474 with an extra existential quantifier. (Contributed by Rohan Ridenour, 12-Aug-2023.)
(𝜑𝑥𝐴)    &   (𝜑 → ∃𝑦 𝑥𝐹𝑦)       (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
 
Theoremgrucollcld 40476 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.)
(𝜑𝐺 ∈ Univ)    &   (𝜑𝐹 ⊆ (𝐺 × 𝐺))    &   (𝜑𝐴𝐺)       (𝜑 → (𝐹 Coll 𝐴) ∈ 𝐺)
 
20.32.2.2  Minimal universes
 
Theoremismnu 40477* The hypothesis of this theorem defines a class M of sets that we temporarily call "minimal universes", and which will turn out in grumnueq 40503 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 40515, except for the 𝑥𝑦 clause.

A minimal universe is closed under subsets (mnussd 40479), powersets (mnupwd 40483), and an operation which is similar to a combination of collection and union (mnuop3d 40487), from which closure under pairing (mnuprd 40492), unions (mnuunid 40493), and function ranges (mnurnd 40499) can be deduced, from which equivalence with Grothendieck universes (grumnueq 40503) can be deduced. (Contributed by Rohan Ridenour, 13-Aug-2023.)

𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}       (𝑈𝑉 → (𝑈𝑀 ↔ ∀𝑧𝑈 (𝒫 𝑧𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))))
 
Theoremmnuop123d 40478* Operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 → (𝒫 𝐴𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
 
Theoremmnussd 40479* Minimal universes are closed under subsets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝐴)       (𝜑𝐵𝑈)
 
Theoremmnuss2d 40480* mnussd 40479 with arguments provided with an existential quantifier. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑 → ∃𝑥𝑈 𝐴𝑥)       (𝜑𝐴𝑈)
 
Theoremmnu0eld 40481* A nonempty minimal universe contains the empty set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 → ∅ ∈ 𝑈)
 
Theoremmnuop23d 40482* Second and third operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹𝑉)       (𝜑 → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
 
Theoremmnupwd 40483* Minimal universes are closed under powersets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 → 𝒫 𝐴𝑈)
 
Theoremmnusnd 40484* Minimal universes are closed under singletons. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 → {𝐴} ∈ 𝑈)
 
Theoremmnuprssd 40485* A minimal universe contains pairs of subsets of an element of the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐶𝑈)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremmnuprss2d 40486* Special case of mnuprssd 40485. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐶𝑈)    &   𝐴𝐶    &   𝐵𝐶       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremmnuop3d 40487* Third operation of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹𝑈)       (𝜑 → ∃𝑤𝑈𝑖𝐴 (∃𝑣𝐹 𝑖𝑣 → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))
 
Theoremmnuprdlem1 40488* Lemma for mnuprd 40492. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))       (𝜑𝐴𝑤)
 
Theoremmnuprdlem2 40489* Lemma for mnuprd 40492. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   (𝜑𝐵𝑈)    &   (𝜑 → ¬ 𝐴 = ∅)    &   (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))       (𝜑𝐵𝑤)
 
Theoremmnuprdlem3 40490* Lemma for mnuprd 40492. (Contributed by Rohan Ridenour, 11-Aug-2023.)
𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   𝑖𝜑       (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑣𝐹 𝑖𝑣)
 
Theoremmnuprdlem4 40491* Lemma for mnuprd 40492. General case. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑 → ¬ 𝐴 = ∅)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremmnuprd 40492* Minimal universes are closed under pairing. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremmnuunid 40493* Minimal universes are closed under union. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)       (𝜑 𝐴𝑈)
 
Theoremmnuund 40494* Minimal universes are closed under binary unions. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremmnutrcld 40495* Minimal universes contain the elements of their elements. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝐴)       (𝜑𝐵𝑈)
 
Theoremmnutrd 40496* Minimal universes are transitive. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)       (𝜑 → Tr 𝑈)
 
Theoremmnurndlem1 40497* Lemma for mnurnd 40499. (Contributed by Rohan Ridenour, 12-Aug-2023.)
(𝜑𝐹:𝐴𝑈)    &   𝐴 ∈ V    &   (𝜑 → ∀𝑖𝐴 (∃𝑣 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})𝑖𝑣 → ∃𝑢 ∈ ran (𝑎𝐴 ↦ {𝑎, {(𝐹𝑎), 𝐴}})(𝑖𝑢 𝑢𝑤)))       (𝜑 → ran 𝐹𝑤)
 
Theoremmnurndlem2 40498* Lemma for mnurnd 40499. Deduction theorem input. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹:𝐴𝑈)    &   𝐴 ∈ V       (𝜑 → ran 𝐹𝑈)
 
Theoremmnurnd 40499* Minimal universes contain ranges of functions from an element of the universe to the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹:𝐴𝑈)       (𝜑 → ran 𝐹𝑈)
 
Theoremmnugrud 40500* Minimal universes are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝑈𝑀)       (𝜑𝑈 ∈ Univ)
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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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