P. 108 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10701-10800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tskpw 10701	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 10702	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 10703	The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)
⊢ ∅ ∈ Tarski
Theorem	tsksdom 10704	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 10705	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 10706	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 10707	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 10708	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 10709	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 10710	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 10711	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 10712	One is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.)
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o ∈ 𝑇)
Theorem	tsk2 10713	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 10714	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 10715	A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 9583.) (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
Theorem	tskr1om2 10716	A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 9583.) (Contributed by NM, 22-Feb-2011.)
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∪ (𝑅1 “ ω) ⊆ 𝑇)
Theorem	tskinf 10717	A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.)
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇)
Theorem	tskpr 10718	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 10719	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 10720	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 10721	A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
⊢ (𝑇 ∈ Tarski → 𝑇 ∈ dom card)
Theorem	inttsk 10722	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 10723	(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 10724	Alternate proof of r1om 10189, shorter as a consequence of inar1 10723, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅1‘ω) ≈ ω
Theorem	rankcf 10725	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 10726	(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 10727	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 10728	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 10729	An even more direct relationship than r1tskina 10730 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 10730	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 10731	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 10732	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 10733	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 10734	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 10735	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 10736	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 10737	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 10738	Extend class notation to include the class of all Grothendieck universes.
class Univ
Definition	df-gru 10739*	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 10740*	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 10741	A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝑈 ∈ Univ → Tr 𝑈)
Theorem	gruelss 10742	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 10743	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 10744	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 10745	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 10746	A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10747 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈)
Theorem	gruiun 10747*	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 10748	A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝐴 ∈ 𝑈)
Theorem	grurn 10749	A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10747 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈)
Theorem	gruima 10750	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 10751	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 10752	A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈)
Theorem	gruop 10753	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 10754	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 10755	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 10756	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 10757*	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 10758*	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 10759	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 10760	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 10761	A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni)
Theorem	intgru 10762	The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ)
Theorem	ingru 10763*	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 10764	The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ)
Theorem	grudomon 10765	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 10766	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 10767	A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 = (𝑈 ∩ On)    ⇒   ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈)
Theorem	grur1 10768	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 10769	Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 10731.) (Contributed by Mario Carneiro, 17-Jun-2013.)
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ)
Theorem	grutsk 10770	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 10771*	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 10782). 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 10772*	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 10773*	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 10774*	Derive the Axiom of Power Sets ax-pow 5316 from the Tarski-Grothendieck axiom ax-groth 10771. That it follows is mentioned by Bob Solovay at https://fomarchive.ugent.be/2008-March/012783.html 10771. Note that ax-pow 5316 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 10775	Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10771. Note that ax-pow 5316 is not used by the proof. Use axpweq 5301 to obtain ax-pow 5316. Use pwex 5331 or pwexg 5329 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
⊢ 𝒫 𝑥 ∈ V
Theorem	axgroth6 10776*	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 10777	The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9588). 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 10778	The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10416). This can be put in a more conventional form via ween 9981 and dfac8 10082. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see https://fomarchive.ugent.be/2008-March/012783.html 10082). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
⊢ dom card = V
Theorem	axgroth3 10779*	Alternate version of the Tarski-Grothendieck Axiom. ax-cc 10382 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 10780*	Alternate version of the Tarski-Grothendieck Axiom. ax-ac 10406 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 10781*	Lemma for grothprim 10782. 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 10782*	The Tarski-Grothendieck Axiom ax-groth 10771 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 10783	The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
⊢ ∪ Tarski = V
Theorem	inaprc 10784	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 10785	Extend class definition to include the map whose value is the smallest Tarski class.
class tarskiMap
Definition	df-tskm 10786*	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 10787*	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 10788	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 10789	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
Theorem	sstskm 10790*	Being a part of (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
Theorem	eltskm 10791*	Belonging to (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))
PART 5  REAL AND COMPLEX NUMBERS This section derives the basics of real and complex numbers. We first construct and axiomatize real and complex numbers (e.g., ax-resscn 11120). After that, we derive their basic properties, various operations like addition (df-add 11074) and sine (df-sin 16075), and subsets such as the integers (df-z 12559) and natural numbers (df-nn 12201).
5.1  Construction and axiomatization of real and complex numbers
5.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 10792	The set of positive integers, which is the set of natural numbers ω with 0 removed. Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 11092. The actual set of Dedekind cuts is defined by df-np 10929.
class N
Syntax	cpli 10793	Positive integer addition.
class +N
Syntax	cmi 10794	Positive integer multiplication.
class ·N
Syntax	clti 10795	Positive integer ordering relation.
class <N
Syntax	cplpq 10796	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 10797	Positive pre-fraction multiplication.
class ·pQ
Syntax	cltpq 10798	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 10799	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 10800	Set of positive fractions.
class Q