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Theorem List for Metamath Proof Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwinainf 9501 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → ω ⊆ 𝐴)

Theoremwinalim 9502 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → Lim 𝐴)

Theoremwinalim2 9503* A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)
((𝐴 ∈ Inaccw𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))

Theoremwinafp 9504 A nontrivial weakly inaccessible cardinal is a fixed point of the aleph function. (Contributed by Mario Carneiro, 29-May-2014.)
((𝐴 ∈ Inaccw𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴)

Theoremwinafpi 9505 This theorem, which states that a nontrivial inaccessible cardinal is its own aleph number, is stated here in inference form, where the assumptions are in the hypotheses rather than an antecedent. Often, we use dedth 4130 to turn this type of statement into the closed form statement winafp 9504, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 9504 using this theorem and dedth 4130, in ZFC. (You can prove this if you use ax-groth 9630, though.) (Contributed by Mario Carneiro, 28-May-2014.)
𝐴 ∈ Inaccw    &   𝐴 ≠ ω       (ℵ‘𝐴) = 𝐴

Theoremgchina 9506 Assuming the GCH, weakly and strongly inaccessible cardinals coincide. Theorem 11.20 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 5-Jun-2015.)
(GCH = V → Inaccw = Inacc)

4.1.2  Weak universes

Syntaxcwun 9507 Extend class definition to include the class of all weak universes.
class WUni

Syntaxcwunm 9508 Extend class definition to include the map whose value is the smallest weak universe of which the given set is a subset.
class wUniCl

Definitiondf-wun 9509* The class of all weak universes. A weak universe is a nonempty transitive class closed under union, pairing, and powerset. The advantage of weak universes over Grothendieck universes is that one can prove that every set is contained in a weak universe in ZF (see uniwun 9547) whereas the analogue for Grothendieck universes requires ax-groth 9630 (see grothtsk 9642). (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = {𝑢 ∣ (Tr 𝑢𝑢 ≠ ∅ ∧ ∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢))}

Definitiondf-wunc 9510* A function that maps a set 𝑥 to the smallest weak universe that contains the elements of the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
wUniCl = (𝑥 ∈ V ↦ {𝑢 ∈ WUni ∣ 𝑥𝑢})

Theoremiswun 9511* Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈𝑉 → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))

Theoremwuntr 9512 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ WUni → Tr 𝑈)

Theoremwununi 9513 A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 𝐴𝑈)

Theoremwunpw 9514 A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → 𝒫 𝐴𝑈)

Theoremwunelss 9515 The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)

Theoremwunpr 9516 A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Theoremwunun 9517 A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwuntp 9518 A weak universe is closed under unordered triple. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑𝐶𝑈)       (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)

Theoremwunss 9519 A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝐴)       (𝜑𝐵𝑈)

Theoremwunin 9520 A weak universe is closed under binary intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwundif 9521 A weak universe is closed under class difference. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwunint 9522 A weak universe is closed under nonempty intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)

Theoremwunsn 9523 A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → {𝐴} ∈ 𝑈)

Theoremwunsuc 9524 A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → suc 𝐴𝑈)

Theoremwun0 9525 A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)       (𝜑 → ∅ ∈ 𝑈)

Theoremwunr1om 9526 A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)       (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)

Theoremwunom 9527 A weak universe contains all the finite ordinals, and hence is infinite. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)       (𝜑 → ω ⊆ 𝑈)

Theoremwunfi 9528 A weak universe contains all finite sets with elements drawn from the universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐴 ∈ Fin)       (𝜑𝐴𝑈)

Theoremwunop 9529 A weak universe is closed under ordered pairs. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Theoremwunot 9530 A weak universe is closed under ordered triples. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑𝐶𝑈)       (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)

Theoremwunxp 9531 A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)

Theoremwunpm 9532 A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴pm 𝐵) ∈ 𝑈)

Theoremwunmap 9533 A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝑚 𝐵) ∈ 𝑈)

Theoremwunf 9534 A weak universe is closed under functions with known domain and codomain. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐹𝑈)

Theoremwundm 9535 A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → dom 𝐴𝑈)

Theoremwunrn 9536 A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → ran 𝐴𝑈)

Theoremwuncnv 9537 A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)

Theoremwunres 9538 A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwunfv 9539 A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwunco 9540 A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwuntpos 9541 A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → tpos 𝐴𝑈)

Theoremintwun 9542 The intersection of a collection of weak universes is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝐴 ⊆ WUni ∧ 𝐴 ≠ ∅) → 𝐴 ∈ WUni)

Theoremr1limwun 9543 Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)

Theoremr1wunlim 9544 The weak universes in the cumulative hierarchy are exactly the limit ordinals. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))

Theoremwunex2 9545* Construct a weak universe from a given set. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω)    &   𝑈 = ran 𝐹       (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))

Theoremwunex 9546* Construct a weak universe from a given set. See also wunex2 9545. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → ∃𝑢 ∈ WUni 𝐴𝑢)

Theoremuniwun 9547 Every set is contained in a weak universe. This is the analogue of grothtsk 9642 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 9642. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = V

Theoremwunex3 9548 Construct a weak universe from a given set. This version of wunex 9546 has a simpler proof, but requires the axiom of regularity. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑈 = (𝑅1‘((rank‘𝐴) +𝑜 ω))       (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))

Theoremwuncval 9549* Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) = {𝑢 ∈ WUni ∣ 𝐴𝑢})

Theoremwuncid 9550 The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))

Theoremwunccl 9551 The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)

Theoremwuncss 9552 The weak universe closure is a subset of any other weak universe containing the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑈 ∈ WUni ∧ 𝐴𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)

Theoremwuncidm 9553 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴))

Theoremwuncval2 9554* Our earlier expression for a containing weak universe is in fact the weak universe closure. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω)    &   𝑈 = ran 𝐹       (𝐴𝑉 → (wUniCl‘𝐴) = 𝑈)

4.1.3  Tarski classes

Syntaxctsk 9555 Extend class definition to include the class of all Tarski classes.
class Tarski

Definitiondf-tsk 9556* The class of all Tarski classes. Tarski classes is a phrase coined by Grzegorz Bancerek in his article Tarski's Classes and Ranks, Journal of Formalized Mathematics, Vol 1, No 3, May-August 1990. A Tarski class is a set whose existence is ensured by Tarski's axiom A (see ax-groth 9630 and the equivalent axioms). Axiom A was first presented in Tarski's article _Über unerreichbare Kardinalzahlen_. Tarski introduced the axiom A to enable ZFC to manage inaccessible cardinals. Later Grothendieck introduced the concept of Grothendieck universes and showed they were equal to transitive Tarski classes. (Contributed by FL, 30-Dec-2010.)
Tarski = {𝑦 ∣ (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))}

Theoremeltskg 9557* Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.)
(𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))

Theoremeltsk2g 9558* Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
(𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))

Theoremtskpwss 9559 First axiom of a Tarski class. The subsets of an element of a Tarski class belong to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)

Theoremtskpw 9560 Second axiom of a Tarski class. The powerset of an element of a Tarski class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)

Theoremtsken 9561 Third axiom of a Tarski class. A subset of a Tarski class is either equipotent to the class or an element of the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))

Theorem0tsk 9562 The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)
∅ ∈ Tarski

Theoremtsksdom 9563 An element of a Tarski class is strictly dominated by the class. JFM CLASSES2 th. 1. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)

Theoremtskssel 9564 A part of a Tarski class strictly dominated by the class is an element of the class. JFM CLASSES2 th. 2. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → 𝐴𝑇)

Theoremtskss 9565 The subsets of an element of a Tarski class belong to the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝐴) → 𝐵𝑇)

Theoremtskin 9566 The intersection of two elements of a Tarski class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝐵) ∈ 𝑇)

Theoremtsksn 9567 A singleton of an element of a Tarski class belongs to the class. JFM CLASSES2 th. 2 (partly). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇) → {𝐴} ∈ 𝑇)

Theoremtsktrss 9568 A transitive element of a Tarski class is a part of the class. JFM CLASSES2 th. 8. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 20-Sep-2014.)
((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴𝑇)

Theoremtsksuc 9569 If an element of a Tarski class is an ordinal number, its successor is an element of the class. JFM CLASSES2 th. 6 (partly). (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → suc 𝐴𝑇)

Theoremtsk0 9570 A nonempty Tarski class contains the empty set. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)

Theoremtsk1 9571 One is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1𝑜𝑇)

Theoremtsk2 9572 Two is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2𝑜𝑇)

Theorem2domtsk 9573 If a Tarski class is not empty, it has more than two elements. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2𝑜𝑇)

Theoremtskr1om 9574 A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 8520.) (Contributed by Mario Carneiro, 24-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Theoremtskr1om2 9575 A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8520.) (Contributed by NM, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Theoremtskinf 9576 A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇)

Theoremtskpr 9577 If 𝐴 and 𝐵 are members of a Tarski class, their unordered pair is also an element of the class. JFM CLASSES2 th. 3 (partly). (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝑇) → {𝐴, 𝐵} ∈ 𝑇)

Theoremtskop 9578 If 𝐴 and 𝐵 are members of a Tarski class, their ordered pair is also an element of the class. JFM CLASSES2 th. 4. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝑇) → ⟨𝐴, 𝐵⟩ ∈ 𝑇)

Theoremtskxpss 9579 A Cartesian product of two parts of a Tarski class is a part of the class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)

Theoremtskwe2 9580 A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
(𝑇 ∈ Tarski → 𝑇 ∈ dom card)

Theoreminttsk 9581 The intersection of a collection of Tarski classes is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝐴 ⊆ Tarski ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Tarski)

Theoreminar1 9582 (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is equipotent to 𝐴. (Contributed by Mario Carneiro, 6-Jun-2013.)
(𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)

Theoremr1omALT 9583 Alternate proof of r1om 9051, shorter as a consequence of inar1 9582, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑅1‘ω) ≈ ω

Theoremrankcf 9584 Any set must be at least as large as the cofinality of its rank, because the ranks of the elements of 𝐴 form a cofinal map into (rank‘𝐴). (Contributed by Mario Carneiro, 27-May-2013.)
¬ 𝐴 ≺ (cf‘(rank‘𝐴))

Theoreminatsk 9585 (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is a Tarski class. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)

Theoremr1omtsk 9586 The set of hereditarily finite sets is a Tarski class. (The Tarski-Grothendieck Axiom is not needed for this theorem.) (Contributed by Mario Carneiro, 28-May-2013.)
(𝑅1‘ω) ∈ Tarski

Theoremtskord 9587 A Tarski class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝐴𝑇)

Theoremtskcard 9588 An even more direct relationship than r1tskina 9589 to get an inaccessible cardinal out of a Tarski class: the size of any nonempty Tarski class is an inaccessible cardinal. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (card‘𝑇) ∈ Inacc)

Theoremr1tskina 9589 There is a direct relationship between transitive Tarski classes and inaccessible cardinals: the Tarski classes that occur in the cumulative hierarchy are exactly at the strongly inaccessible cardinals. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 ∈ On → ((𝑅1𝐴) ∈ Tarski ↔ (𝐴 = ∅ ∨ 𝐴 ∈ Inacc)))

Theoremtskuni 9590 The union of an element of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 22-Jun-2013.)
((𝑇 ∈ Tarski ∧ Tr 𝑇𝐴𝑇) → 𝐴𝑇)

Theoremtskwun 9591 A nonempty transitive Tarski class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑇 ∈ Tarski ∧ Tr 𝑇𝑇 ≠ ∅) → 𝑇 ∈ WUni)

Theoremtskint 9592 The intersection of an element of a transitive Tarski class is an element of the class. (Contributed by FL, 17-Apr-2011.) (Revised by Mario Carneiro, 20-Sep-2014.)
(((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐴 ≠ ∅) → 𝐴𝑇)

Theoremtskun 9593 The union of two elements of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 20-Sep-2014.)
(((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐵𝑇) → (𝐴𝐵) ∈ 𝑇)

Theoremtskxp 9594 The Cartesian product of two elements of a transitive Tarski class is an element of the class. JFM CLASSES2 th. 67 (partly). (Contributed by FL, 15-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
(((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ∈ 𝑇)

Theoremtskmap 9595 Set exponentiation is an element of a transitive Tarski class. JFM CLASSES2 th. 67 (partly). (Contributed by FL, 15-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
(((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐵𝑇) → (𝐴𝑚 𝐵) ∈ 𝑇)

Theoremtskurn 9596 A transitive Tarski class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.)
(((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)

4.1.4  Grothendieck universes

Syntaxcgru 9597 Extend class notation to include the class of all Grothendieck universes.
class Univ

Definitiondf-gru 9598* A Grothendieck universe is a set that is closed with respect to all the operations that are common in set theory: pairs, powersets, unions, intersections, Cartesian products etc. Grothendieck and alii, Séminaire de Géométrie Algébrique 4, Exposé I, p. 185. It was designed to give a precise meaning to the concepts of categories of sets, groups... (Contributed by Mario Carneiro, 9-Jun-2013.)
Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢))}

Theoremelgrug 9599* Properties of a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
(𝑈𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈))))

Theoremgrutr 9600 A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ Univ → Tr 𝑈)

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