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Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwunex3 9601 Construct a weak universe from a given set. This version of wunex 9599 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 9602* Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) = {𝑢 ∈ WUni ∣ 𝐴𝑢})

Theoremwuncid 9603 The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))

Theoremwunccl 9604 The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)

Theoremwuncss 9605 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 9606 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴))

Theoremwuncval2 9607* 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 9608 Extend class definition to include the class of all Tarski classes.
class Tarski

Definitiondf-tsk 9609* 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 9683 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 9610* 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 9611* 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 9612 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 9613 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 9614 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 9615 The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)
∅ ∈ Tarski

Theoremtsksdom 9616 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 9617 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 9618 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 9619 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 9620 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 9621 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 9622 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 9623 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 9624 One is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1𝑜𝑇)

Theoremtsk2 9625 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 9626 If a Tarski class is not empty, it has more than two elements. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2𝑜𝑇)

Theoremtskr1om 9627 A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 8573.) (Contributed by Mario Carneiro, 24-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Theoremtskr1om2 9628 A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8573.) (Contributed by NM, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)

Theoremtskinf 9629 A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇)

Theoremtskpr 9630 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 9631 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 9632 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 9633 A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
(𝑇 ∈ Tarski → 𝑇 ∈ dom card)

Theoreminttsk 9634 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 9635 (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is equipotent to 𝐴. (Contributed by Mario Carneiro, 6-Jun-2013.)
(𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)

Theoremr1omALT 9636 Alternate proof of r1om 9104, shorter as a consequence of inar1 9635, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑅1‘ω) ≈ ω

Theoremrankcf 9637 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 9638 (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is a Tarski class. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)

Theoremr1omtsk 9639 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 9640 A Tarski class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝐴𝑇)

Theoremtskcard 9641 An even more direct relationship than r1tskina 9642 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 9642 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 9643 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 9644 A nonempty transitive Tarski class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑇 ∈ Tarski ∧ Tr 𝑇𝑇 ≠ ∅) → 𝑇 ∈ WUni)

Theoremtskint 9645 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 9646 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 9647 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 9648 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 9649 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 9650 Extend class notation to include the class of all Grothendieck universes.
class Univ

Definitiondf-gru 9651* 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 9652* 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 9653 A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ Univ → Tr 𝑈)

Theoremgruelss 9654 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 9655 A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝒫 𝐴𝑈)

Theoremgruss 9656 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 9657 A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)

Theoremgruurn 9658 A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9659 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)

Theoremgruiun 9659* 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 9660 A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)

Theoremgrurn 9661 A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9659 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)

Theoremgruima 9662 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 9663 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 9664 A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)

Theoremgruop 9665 A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Theoremgruun 9666 A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)

Theoremgruxp 9667 A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴 × 𝐵) ∈ 𝑈)

Theoremgrumap 9668 A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝑚 𝐵) ∈ 𝑈)

Theoremgruixp 9669* 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 9670* A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝑈 ∈ Univ ∧ ∃𝑥𝐴 𝐵𝑈) → 𝑥𝐴 𝐵𝑈)

Theoremgruf 9671 A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → 𝐹𝑈)

Theoremgruen 9672 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 9673 A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni)

Theoremintgru 9674 The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)

Theoremingru 9675* 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 9676 The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
(𝑈 ∈ Univ → (𝑈 (𝑅1 “ On)) ∈ Univ)

Theoremgrudomon 9677 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 9678 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 9679 A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 = (𝑈 ∩ On)       (𝑈 ∈ Univ → (𝑅1𝐴) ⊆ 𝑈)

Theoremgrur1 9680 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 9681 Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 9643.) (Contributed by Mario Carneiro, 17-Jun-2013.)
((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)

Theoremgrutsk 9682 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 9683* 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 9694. An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))

Theoremaxgroth5 9684* The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))

Theoremaxgroth2 9685* Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑦𝑧𝑧𝑦)))

4.2.2  Derive the Power Set, Infinity and Choice Axioms

Theoremgrothpw 9686* Derive the Axiom of Power Sets ax-pow 4873 from the Tarski-Grothendieck axiom ax-groth 9683. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4873 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)

Theoremgrothpwex 9687 Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 9683. Note that ax-pow 4873 is not used by the proof. Use axpweq 4872 to obtain ax-pow 4873. Use pwex 4878 or pwexg 4880 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
𝒫 𝑥 ∈ V

Theoremaxgroth6 9688* The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set 𝑥, there exists a set 𝑦 containing 𝑥, the subsets of the members of 𝑦, the power sets of the members of 𝑦, and the subsets of 𝑦 of cardinality less than that of 𝑦. (Contributed by NM, 21-Jun-2009.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))

Theoremgrothomex 9689 The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 8578). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
ω ∈ V

Theoremgrothac 9690 The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9329). This can be put in a more conventional form via ween 8896 and dfac8 8995. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
dom card = V

Theoremaxgroth3 9691* Alternate version of the Tarski-Grothendieck Axiom. ax-cc 9295 is used to derive this version. (Contributed by NM, 26-Mar-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))

Theoremaxgroth4 9692* Alternate version of the Tarski-Grothendieck Axiom. ax-ac 9319 is used to derive this version. (Contributed by NM, 16-Apr-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦𝑣𝑦𝑤(𝑤𝑧𝑤 ∈ (𝑦𝑣)) ∧ ∀𝑧(𝑧𝑦 → ((𝑦𝑧) ≼ 𝑧𝑧𝑦)))

Theoremgrothprimlem 9693* Lemma for grothprim 9694. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))

Theoremgrothprim 9694* The Tarski-Grothendieck Axiom ax-groth 9683 expanded into set theory primitives using 163 symbols (allowing the defined symbols , , , and ). An open problem is whether a shorter equivalent exists (when expanded to primitives). (Contributed by NM, 16-Apr-2007.)
𝑦(𝑥𝑦 ∧ ∀𝑧((𝑧𝑦 → ∃𝑣(𝑣𝑦 ∧ ∀𝑤(∀𝑢(𝑢𝑤𝑢𝑧) → (𝑤𝑦𝑤𝑣)))) ∧ ∃𝑤((𝑤𝑧𝑤𝑦) → (∀𝑣((𝑣𝑧 → ∃𝑡𝑢(∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑣 = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣𝑦 → (𝑣𝑧 ∨ ∃𝑢(𝑢𝑧 ∧ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))))) ∨ 𝑧𝑦))))

Theoremgrothtsk 9695 The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
Tarski = V

Theoreminaprc 9696 An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.)
Inacc ∉ V

4.2.3  Tarski map function

Syntaxctskm 9697 Extend class definition to include the map whose value is the smallest Tarski class.
class tarskiMap

Definitiondf-tskm 9698* A function that maps a set 𝑥 to the smallest Tarski class that contains the set. (Contributed by FL, 30-Dec-2010.)
tarskiMap = (𝑥 ∈ V ↦ {𝑦 ∈ Tarski ∣ 𝑥𝑦})

Theoremtskmval 9699* Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
(𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})

Theoremtskmid 9700 The set 𝐴 is an element of the smallest Tarski class that contains 𝐴. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
(𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))

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