P. 100 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wuncval 9901*	Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢})
Theorem	wuncid 9902	The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (wUniCl‘𝐴))
Theorem	wunccl 9903	The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ∈ WUni)
Theorem	wuncss 9904	The weak universe closure is a subset of any other weak universe containing the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ ((𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)
Theorem	wuncidm 9905	The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝐴 ∈ 𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴))
Theorem	wuncval2 9906*	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) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)    &   ⊢ 𝑈 = ∪ ran 𝐹    ⇒   ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = 𝑈)
4.1.3  Tarski classes
Syntax	ctsk 9907	Extend class definition to include the class of all Tarski classes.
class Tarski
Definition	df-tsk 9908*	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 9982 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 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
Theorem	eltskg 9909*	Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.)
⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))
Theorem	eltsk2g 9910*	Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))
Theorem	tskpwss 9911	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 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇)
Theorem	tskpw 9912	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 ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
Theorem	tsken 9913	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 ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
Theorem	0tsk 9914	The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)
⊢ ∅ ∈ Tarski
Theorem	tsksdom 9915	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 ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ 𝑇)
Theorem	tskssel 9916	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 ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)
Theorem	tskss 9917	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 ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑇)
Theorem	tskin 9918	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 ∧ 𝐴 ∈ 𝑇) → (𝐴 ∩ 𝐵) ∈ 𝑇)
Theorem	tsksn 9919	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 ∧ 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇)
Theorem	tsktrss 9920	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 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)
Theorem	tsksuc 9921	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 ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ∈ 𝑇)
Theorem	tsk0 9922	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)
Theorem	tsk1 9923	One is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.)
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o ∈ 𝑇)
Theorem	tsk2 9924	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 ≠ ∅) → 2o ∈ 𝑇)
Theorem	2domtsk 9925	If a Tarski class is not empty, it has more than two elements. (Contributed by FL, 22-Feb-2011.)
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2o ≺ 𝑇)
Theorem	tskr1om 9926	A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 8834.) (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
Theorem	tskr1om2 9927	A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8834.) (Contributed by NM, 22-Feb-2011.)
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∪ (𝑅1 “ ω) ⊆ 𝑇)
Theorem	tskinf 9928	A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.)
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇)
Theorem	tskpr 9929	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 ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ∈ 𝑇)
Theorem	tskop 9930	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 ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → ⟨𝐴, 𝐵⟩ ∈ 𝑇)
Theorem	tskxpss 9931	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 ∧ 𝐴 ⊆ 𝑇 ∧ 𝐵 ⊆ 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
Theorem	tskwe2 9932	A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
⊢ (𝑇 ∈ Tarski → 𝑇 ∈ dom card)
Theorem	inttsk 9933	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)
Theorem	inar1 9934	(𝑅1‘𝐴) for 𝐴 a strongly inaccessible cardinal is equipotent to 𝐴. (Contributed by Mario Carneiro, 6-Jun-2013.)
⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ≈ 𝐴)
Theorem	r1omALT 9935	Alternate proof of r1om 9403, shorter as a consequence of inar1 9934, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅1‘ω) ≈ ω
Theorem	rankcf 9936	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))
Theorem	inatsk 9937	(𝑅1‘𝐴) for 𝐴 a strongly inaccessible cardinal is a Tarski class. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ∈ Tarski)
Theorem	r1omtsk 9938	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
Theorem	tskord 9939	A Tarski class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)
Theorem	tskcard 9940	An even more direct relationship than r1tskina 9941 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)
Theorem	r1tskina 9941	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 → ((𝑅1‘𝐴) ∈ Tarski ↔ (𝐴 = ∅ ∨ 𝐴 ∈ Inacc)))
Theorem	tskuni 9942	The union of an element of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 22-Jun-2013.)
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ∈ 𝑇)
Theorem	tskwun 9943	A nonempty transitive Tarski class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → 𝑇 ∈ WUni)
Theorem	tskint 9944	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) ∧ 𝐴 ∈ 𝑇 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑇)
Theorem	tskun 9945	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) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ∪ 𝐵) ∈ 𝑇)
Theorem	tskxp 9946	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) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 × 𝐵) ∈ 𝑇)
Theorem	tskmap 9947	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) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ↑𝑚 𝐵) ∈ 𝑇)
Theorem	tskurn 9948	A transitive Tarski class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.)
⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹:𝐴⟶𝑇) → ∪ ran 𝐹 ∈ 𝑇)
4.1.4  Grothendieck universes
Syntax	cgru 9949	Extend class notation to include the class of all Grothendieck universes.
class Univ
Definition	df-gru 9950*	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 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑢))}
Theorem	elgrug 9951*	Properties of a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈))))
Theorem	grutr 9952	A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝑈 ∈ Univ → Tr 𝑈)
Theorem	gruelss 9953	A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈)
Theorem	grupw 9954	A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈)
Theorem	gruss 9955	Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑈)
Theorem	grupr 9956	A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
Theorem	gruurn 9957	A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9958 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈)
Theorem	gruiun 9958*	If 𝐵(𝑥) is a family of elements of 𝑈 and the index set 𝐴 is an element of 𝑈, then the indexed union ∪ 𝑥 ∈ 𝐴𝐵 is also an element of 𝑈, where 𝑈 is a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
Theorem	gruuni 9959	A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝐴 ∈ 𝑈)
Theorem	grurn 9960	A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9958 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈)
Theorem	gruima 9961	A Grothendieck universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈))
Theorem	gruel 9962	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 ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝑈)
Theorem	grusn 9963	A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈)
Theorem	gruop 9964	A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
Theorem	gruun 9965	A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈)
Theorem	gruxp 9966	A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈)
Theorem	grumap 9967	A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ↑𝑚 𝐵) ∈ 𝑈)
Theorem	gruixp 9968*	A Grothendieck universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
Theorem	gruiin 9969*	A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
Theorem	gruf 9970	A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝐹 ∈ 𝑈)
Theorem	gruen 9971	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 ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈)
Theorem	gruwun 9972	A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni)
Theorem	intgru 9973	The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ)
Theorem	ingru 9974*	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 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑈 ∈ Univ → (𝑈 ∩ 𝐴) ∈ Univ))
Theorem	wfgru 9975	The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ)
Theorem	grudomon 9976	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 ∧ (𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵)) → 𝐴 ∈ 𝑈)
Theorem	gruina 9977	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)
Theorem	grur1a 9978	A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 = (𝑈 ∩ On)    ⇒   ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈)
Theorem	grur1 9979	A characterization of Grothendieck universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝐴 = (𝑈 ∩ On)    ⇒   ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 = (𝑅1‘𝐴))
Theorem	grutsk1 9980	Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 9942.) (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
Theorem	grutsk 9981	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
Axiom	ax-groth 9982*	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 9993. An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
Theorem	axgroth5 9983*	The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))
Theorem	axgroth2 9984*	Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)))
4.2.2  Derive the Power Set, Infinity and Choice Axioms
Theorem	grothpw 9985*	Derive the Axiom of Power Sets ax-pow 5079 from the Tarski-Grothendieck axiom ax-groth 9982. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 5079 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	grothpwex 9986	Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 9982. Note that ax-pow 5079 is not used by the proof. Use axpweq 5078 to obtain ax-pow 5079. Use pwex 5094 or pwexg 5092 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
⊢ 𝒫 𝑥 ∈ V
Theorem	axgroth6 9987*	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(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))
Theorem	grothomex 9988	The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 8839). 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
Theorem	grothac 9989	The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 9628). This can be put in a more conventional form via ween 9193 and dfac8 9294. 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
Theorem	axgroth3 9990*	Alternate version of the Tarski-Grothendieck Axiom. ax-cc 9594 is used to derive this version. (Contributed by NM, 26-Mar-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)))
Theorem	axgroth4 9991*	Alternate version of the Tarski-Grothendieck Axiom. ax-ac 9618 is used to derive this version. (Contributed by NM, 16-Apr-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∃𝑣 ∈ 𝑦 ∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ (𝑦 ∩ 𝑣)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)))
Theorem	grothprimlem 9992*	Lemma for grothprim 9993. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))
Theorem	grothprim 9993*	The Tarski-Grothendieck Axiom ax-groth 9982 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(𝑥 ∈ 𝑦 ∧ ∀𝑧((𝑧 ∈ 𝑦 → ∃𝑣(𝑣 ∈ 𝑦 ∧ ∀𝑤(∀𝑢(𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑧) → (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝑣)))) ∧ ∃𝑤((𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) → (∀𝑣((𝑣 ∈ 𝑧 → ∃𝑡∀𝑢(∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑣 ∨ ℎ = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣 ∈ 𝑦 → (𝑣 ∈ 𝑧 ∨ ∃𝑢(𝑢 ∈ 𝑧 ∧ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))))) ∨ 𝑧 ∈ 𝑦))))
Theorem	grothtsk 9994	The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
⊢ ∪ Tarski = V
Theorem	inaprc 9995	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
Syntax	ctskm 9996	Extend class definition to include the map whose value is the smallest Tarski class.
class tarskiMap
Definition	df-tskm 9997*	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 ∣ 𝑥 ∈ 𝑦})
Theorem	tskmval 9998*	Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
Theorem	tskmid 9999	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.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))
Theorem	tskmcl 10000	A Tarski class that contains 𝐴 is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
⊢ (tarskiMap‘𝐴) ∈ Tarski