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Theorem List for Metamath Proof Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwunxp 9301 A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)
 
Theoremwunpm 9302 A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴pm 𝐵) ∈ 𝑈)
 
Theoremwunmap 9303 A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝑚 𝐵) ∈ 𝑈)
 
Theoremwunf 9304 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 9305 A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → dom 𝐴𝑈)
 
Theoremwunrn 9306 A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → ran 𝐴𝑈)
 
Theoremwuncnv 9307 A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)
 
Theoremwunres 9308 A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwunfv 9309 A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwunco 9310 A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwuntpos 9311 A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → tpos 𝐴𝑈)
 
Theoremintwun 9312 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 9313 Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
 
Theoremr1wunlim 9314 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 9315* 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 9316* Construct a weak universe from a given set. See also wunex2 9315. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → ∃𝑢 ∈ WUni 𝐴𝑢)
 
Theoremuniwun 9317 Every set is contained in a weak universe. This is the analogue of grothtsk 9412 for weak universes, but it is provable in ZFC without the Tarski-Grothendieck axiom, contrary to grothtsk 9412. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = V
 
Theoremwunex3 9318 Construct a weak universe from a given set. This version of wunex 9316 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 9319* Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) = {𝑢 ∈ WUni ∣ 𝐴𝑢})
 
Theoremwuncid 9320 The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))
 
Theoremwunccl 9321 The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)
 
Theoremwuncss 9322 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 9323 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴))
 
Theoremwuncval2 9324* 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 9325 Extend class definition to include the class of all Tarski classes.
class Tarski
 
Definitiondf-tsk 9326* 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 9400 and the equivalent axioms). Axiom A was first presented in Tarski's article _Über unerreichbare Kardinalzahlen_. Tarski had invented the axiom A to enable ZFC to manage inaccessible cardinals. Later Grothendieck invented 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 9327* 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 9328* 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 9329 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 9330 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 9331 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 9332 The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)
∅ ∈ Tarski
 
Theoremtsksdom 9333 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 9334 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 9335 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 9336 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 9337 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 9338 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 9339 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 9340 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 9341 One is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1𝑜𝑇)
 
Theoremtsk2 9342 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 9343 If a Tarski class is not empty, it has more than two elements. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2𝑜𝑇)
 
Theoremtskr1om 9344 A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 8294.) (Contributed by Mario Carneiro, 24-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
 
Theoremtskr1om2 9345 A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8294.) (Contributed by NM, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
 
Theoremtskinf 9346 A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇)
 
Theoremtskpr 9347 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 9348 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 9349 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 9350 A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
(𝑇 ∈ Tarski → 𝑇 ∈ dom card)
 
Theoreminttsk 9351 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 9352 (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is equipotent to 𝐴. (Contributed by Mario Carneiro, 6-Jun-2013.)
(𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)
 
Theoremr1omALT 9353 Alternate proof of r1om 8825, shorter as a consequence of inar1 9352, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑅1‘ω) ≈ ω
 
Theoremrankcf 9354 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 9355 (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is a Tarski class. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)
 
Theoremr1omtsk 9356 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 9357 A Tarski class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝐴𝑇)
 
Theoremtskcard 9358 An even more direct relationship than r1tskina 9359 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 9359 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 9360 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 9361 A nonempty transitive Tarski class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑇 ∈ Tarski ∧ Tr 𝑇𝑇 ≠ ∅) → 𝑇 ∈ WUni)
 
Theoremtskint 9362 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 9363 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 9364 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 9365 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 9366 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 9367 Extend class notation to include the class of all Grothendieck universes.
class Univ
 
Definitiondf-gru 9368* 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 9369* 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 9370 A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ Univ → Tr 𝑈)
 
Theoremgruelss 9371 A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
 
Theoremgrupw 9372 A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)
 
Theoremgruss 9373 Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝐴) → 𝐵𝑈)
 
Theoremgrupr 9374 A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremgruurn 9375 A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9376 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
 
Theoremgruiun 9376* If 𝐵(𝑥) is a family of elements of 𝑈 and the index set 𝐴 is an element of 𝑈, then the indexed union 𝑥𝐴𝐵 is also an element of 𝑈, where 𝑈 is a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
 
Theoremgruuni 9377 A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
 
Theoremgrurn 9378 A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9376 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
 
Theoremgruima 9379 A Grothendieck universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) → (𝐴𝑈 → (𝐹𝐴) ∈ 𝑈))
 
Theoremgruel 9380 Any element of an element of a Grothendieck universe is also an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝐴) → 𝐵𝑈)
 
Theoremgrusn 9381 A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)
 
Theoremgruop 9382 A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
 
Theoremgruun 9383 A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)
 
Theoremgruxp 9384 A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
 
Theoremgrumap 9385 A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝑚 𝐵) ∈ 𝑈)
 
Theoremgruixp 9386* A Grothendieck universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝐵𝑈) → X𝑥𝐴 𝐵𝑈)
 
Theoremgruiin 9387* A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ ∃𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)
 
Theoremgruf 9388 A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹𝑈)
 
Theoremgruen 9389 A Grothendieck universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (𝐵𝑈𝐵𝐴)) → 𝐴𝑈)
 
Theoremgruwun 9390 A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni)
 
Theoremintgru 9391 The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)
 
Theoremingru 9392* The intersection of a universe with a class that acts like a universe is another universe. (Contributed by Mario Carneiro, 10-Jun-2013.)
((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑈 ∈ Univ → (𝑈𝐴) ∈ Univ))
 
Theoremwfgru 9393 The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
(𝑈 ∈ Univ → (𝑈 (𝑅1 “ On)) ∈ Univ)
 
Theoremgrudomon 9394 Each ordinal that is comparable with an element of the universe is in the universe. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝐵𝑈𝐴𝐵)) → 𝐴𝑈)
 
Theoremgruina 9395 If a Grothendieck universe 𝑈 is nonempty, then the height of the ordinals in 𝑈 is a strongly inaccessible cardinal. (Contributed by Mario Carneiro, 17-Jun-2013.)
𝐴 = (𝑈 ∩ On)       ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc)
 
Theoremgrur1a 9396 A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 = (𝑈 ∩ On)       (𝑈 ∈ Univ → (𝑅1𝐴) ⊆ 𝑈)
 
Theoremgrur1 9397 A characterization of Grothendieck universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.)
𝐴 = (𝑈 ∩ On)       ((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 = (𝑅1𝐴))
 
Theoremgrutsk1 9398 Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 9360.) (Contributed by Mario Carneiro, 17-Jun-2013.)
((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
 
Theoremgrutsk 9399 Grothendieck universes are the same as transitive Tarski classes. (The proof in the forward direction requires Foundation.) (Contributed by Mario Carneiro, 24-Jun-2013.)
Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥}
 
4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
 
4.2.1  Introduce the Tarski-Grothendieck Axiom
 
Axiomax-groth 9400* The Tarski-Grothendieck Axiom. For every set 𝑥 there is an inaccessible cardinal 𝑦 such that 𝑦 is not in 𝑥. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics." This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols - see grothprim 9411. An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
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