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Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwunmap 8601 A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   =>    |-  ( ph  ->  ( A  ^m  B )  e.  U )
 
Theoremwunf 8602 A weak universe is closed under functions with known domain and codomain. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  e.  U )
 
Theoremwundm 8603 A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  dom 
 A  e.  U )
 
Theoremwunrn 8604 A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ran 
 A  e.  U )
 
Theoremwuncnv 8605 A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  `' A  e.  U )
 
Theoremwunres 8606 A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ( A  |`  B )  e.  U )
 
Theoremwunfv 8607 A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ( A `  B )  e.  U )
 
Theoremwunco 8608 A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   =>    |-  ( ph  ->  ( A  o.  B )  e.  U )
 
Theoremwuntpos 8609 A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  -> tpos  A  e.  U )
 
Theoremintwun 8610 The intersection of a collection of weak universes is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ( A  C_ WUni  /\  A  =/=  (/) )  ->  |^| A  e. WUni )
 
Theoremr1limwun 8611 Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ( A  e.  V  /\  Lim  A )  ->  ( R1 `  A )  e. WUni )
 
Theoremr1wunlim 8612 The weak universes in the cumulative hierarchy are exactly the limit ordinals. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  ( ( R1 `  A )  e. WUni  <->  Lim  A ) )
 
Theoremwunex2 8613* Construct a weak universe from a given set. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  F  =  ( rec ( ( z  e. 
 _V  |->  ( ( z  u.  U. z )  u.  U_ x  e.  z  ( { ~P x ,  U. x }  u.  ran  ( y  e.  z  |->  { x ,  y } ) ) ) ) ,  ( A  u.  1o ) )  |`  om )   &    |-  U  =  U. ran  F   =>    |-  ( A  e.  V  ->  ( U  e. WUni  /\  A  C_  U ) )
 
Theoremwunex 8614* Construct a weak universe from a given set. See also wunex2 8613. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  E. u  e. WUni  A  C_  u )
 
Theoremuniwun 8615 Every set is contained in a weak universe. This is the analogue of grothtsk 8710, but it is provable in ZFC without the Tarski-Grothendieck axiom. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 U.WUni  =  _V
 
Theoremwunex3 8616 Construct a weak universe from a given set. This version of wunex 8614 has a simpler proof, but requires the axiom of regularity. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  U  =  ( R1
 `  ( ( rank `  A )  +o  om ) )   =>    |-  ( A  e.  V  ->  ( U  e. WUni  /\  A  C_  U ) )
 
Theoremwuncval 8617* Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  (wUniCl `  A )  =  |^| { u  e. WUni  |  A  C_  u }
 )
 
Theoremwuncid 8618 The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  A  C_  (wUniCl `  A ) )
 
Theoremwunccl 8619 The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  (wUniCl `  A )  e. WUni )
 
Theoremwuncss 8620 The weak universe closure is a subset of any other weak universe containing the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ( U  e. WUni  /\  A  C_  U )  ->  (wUniCl `  A )  C_  U )
 
Theoremwuncidm 8621 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  (wUniCl `  (wUniCl `  A ) )  =  (wUniCl `  A ) )
 
Theoremwuncval2 8622* 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  e. 
 _V  |->  ( ( z  u.  U. z )  u.  U_ x  e.  z  ( { ~P x ,  U. x }  u.  ran  ( y  e.  z  |->  { x ,  y } ) ) ) ) ,  ( A  u.  1o ) )  |`  om )   &    |-  U  =  U. ran  F   =>    |-  ( A  e.  V  ->  (wUniCl `  A )  =  U )
 
4.1.3  Tarski's classes
 
Syntaxctsk 8623 Extend class definition to include the class of all Tarski's classes.
 class  Tarski
 
Definitiondf-tsk 8624* The class of all Tarski's classes. Tarski's 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's class is a set whose existence is ensured by Tarski's axiom A (see ax-groth 8698 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's universes and showed they were equal to transitive Tarski's classes. (Contributed by FL, 30-Dec-2010.)
 |-  Tarski  =  { y  |  ( A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
 ( z  ~~  y  \/  z  e.  y
 ) ) }
 
Theoremeltskg 8625* Properties of a Tarski's class. (Contributed by FL, 30-Dec-2010.)
 |-  ( T  e.  V  ->  ( T  e.  Tarski  <->  ( A. z  e.  T  ( ~P z  C_  T  /\  E. w  e.  T  ~P z  C_  w ) 
 /\  A. z  e.  ~P  T ( z  ~~  T  \/  z  e.  T ) ) ) )
 
Theoremeltsk2g 8626* Properties of a Tarski's class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
 |-  ( T  e.  V  ->  ( T  e.  Tarski  <->  ( A. z  e.  T  ( ~P z  C_  T  /\  ~P z  e.  T )  /\  A. z  e. 
 ~P  T ( z 
 ~~  T  \/  z  e.  T ) ) ) )
 
Theoremtskpwss 8627 1st axiom of a Tarski's class. The subsets of an element of a Tarski's class belong to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( T  e.  Tarski  /\  A  e.  T ) 
 ->  ~P A  C_  T )
 
Theoremtskpw 8628 2nd axiom of a Tarski's class. The powerset of an element of a Tarski's class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( T  e.  Tarski  /\  A  e.  T ) 
 ->  ~P A  e.  T )
 
Theoremtsken 8629 3rd axiom of a Tarski's class. A subset of a Tarski's 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  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
 
Theorem0tsk 8630 The empty set is a (transitive) Tarski's class. (Contributed by FL, 30-Dec-2010.)
 |-  (/)  e.  Tarski
 
Theoremtsksdom 8631 An element of a Tarski's 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  e.  Tarski  /\  A  e.  T ) 
 ->  A  ~<  T )
 
Theoremtskssel 8632 A part of a Tarski's 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  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  A  e.  T )
 
Theoremtskss 8633 The subsets of an element of a Tarski's class belong to the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  C_  A )  ->  B  e.  T )
 
Theoremtskin 8634 The intersection of two elements of a Tarski's class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( T  e.  Tarski  /\  A  e.  T ) 
 ->  ( A  i^i  B )  e.  T )
 
Theoremtsksn 8635 A singleton of an element of a Tarski's class belongs to the class. JFM CLASSES2 th. 2 (partly) (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  A  e.  T ) 
 ->  { A }  e.  T )
 
Theoremtsktrss 8636 A transitive element of a Tarski's 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  e.  Tarski  /\ 
 Tr  A  /\  A  e.  T )  ->  A  C_  T )
 
Theoremtsksuc 8637 If an element of a Tarski's 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  e.  Tarski  /\  A  e.  On  /\  A  e.  T )  ->  suc  A  e.  T )
 
Theoremtsk0 8638 A non-empty Tarski's class contains the empty set. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/) 
 e.  T )
 
Theoremtsk1 8639 One is an element of a non-empty Tarski's class. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  1o  e.  T )
 
Theoremtsk2 8640 Two is an element of a non-empty Tarski's class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  2o  e.  T )
 
Theorem2domtsk 8641 If a Tarski's class is not empty, it has more than two elements. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  2o  ~<  T )
 
Theoremtskr1om 8642 A nonempty Tarski's class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 7593.) (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
 
Theoremtskr1om2 8643 A nonempty Tarski's class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 7593.) (Contributed by NM, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( R1 " om )  C_  T )
 
Theoremtskinf 8644 A nonempty Tarski's class is infinite. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  om 
 ~<_  T )
 
Theoremtskpr 8645 If  A and  B are members of a Tarski's 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  e.  Tarski  /\  A  e.  T  /\  B  e.  T )  ->  { A ,  B }  e.  T )
 
Theoremtskop 8646 If  A and  B are members of a Tarski's class, their ordered pair is also an element of the class. JFM CLASSES2 th. 4. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  e.  T )  -> 
 <. A ,  B >.  e.  T )
 
Theoremtskxpss 8647 A cross product of two parts of a Tarski's class is a part of the class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  B  C_  T )  ->  ( A  X.  B ) 
 C_  T )
 
Theoremtskwe2 8648 A Tarski's class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
 |-  ( T  e.  Tarski  ->  T  e.  dom  card )
 
Theoreminttsk 8649 The intersection of a collection of Tarski's classes is a Tarski's class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( A  C_  Tarski  /\  A  =/=  (/) )  ->  |^| A  e.  Tarski )
 
Theoreminar1 8650  ( R1 `  A ) for  A a strongly inaccessible cardinal is equipotent to  A. (Contributed by Mario Carneiro, 6-Jun-2013.)
 |-  ( A  e.  Inacc  ->  ( R1 `  A ) 
 ~~  A )
 
Theoremr1omALT 8651 The set of hereditarily finite sets is countable. This is a short proof as a consequence of inar1 8650, which requires AC. See r1om 8124 for a direct proof not requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( R1 `  om )  ~~  om
 
Theoremrankcf 8652 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 8653  ( R1 `  A ) for  A a strongly inaccessible cardinal is a Tarski's class. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |-  ( A  e.  Inacc  ->  ( R1 `  A )  e.  Tarski )
 
Theoremr1omtsk 8654 The set of hereditarily finite sets is a Tarski's class. (The Tarski-Grothendieck Axiom is not needed for this theorem.) (Contributed by Mario Carneiro, 28-May-2013.)
 |-  ( R1 `  om )  e.  Tarski
 
Theoremtskord 8655 A Tarski's class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  A  e.  On  /\  A  ~<  T )  ->  A  e.  T )
 
Theoremtskcard 8656 An even more direct relationship than r1tskina 8657 to get an inacessible cardinal out of a Tarski's class: the size of any nonempty Tarski's class is an inaccessible cardinal. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( card `  T )  e.  Inacc )
 
Theoremr1tskina 8657 There is a direct relationship between transitive Tarski's classes and inacessible cardinals: the Tarski's classes that occur in the cumulative hierarchy are exactly at the strongly inaccessible cardinals. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |-  ( A  e.  On  ->  ( ( R1 `  A )  e.  Tarski  <->  ( A  =  (/) 
 \/  A  e.  Inacc ) ) )
 
Theoremtskuni 8658 The union of an element of a transitive Tarski's class is in the set. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\ 
 Tr  T  /\  A  e.  T )  ->  U. A  e.  T )
 
Theoremtskwun 8659 A nonempty transitive Tarski's class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ( T  e.  Tarski  /\ 
 Tr  T  /\  T  =/= 
 (/) )  ->  T  e. WUni )
 
Theoremtskint 8660 The intersection of an element of a transitive Tarski's class is an element of the class. (Contributed by FL, 17-Apr-2011.) (Revised by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( ( T  e.  Tarski  /\  Tr  T ) 
 /\  A  e.  T  /\  A  =/=  (/) )  ->  |^| A  e.  T )
 
Theoremtskun 8661 The union of two elements of a transitive Tarski's class is in the set. (Contributed by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( ( T  e.  Tarski  /\  Tr  T ) 
 /\  A  e.  T  /\  B  e.  T ) 
 ->  ( A  u.  B )  e.  T )
 
Theoremtskxp 8662 The cross product of two elements of a transitive Tarski's 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  e.  Tarski  /\  Tr  T ) 
 /\  A  e.  T  /\  B  e.  T ) 
 ->  ( A  X.  B )  e.  T )
 
Theoremtskmap 8663 Set exponentiation is an element of a transitive Tarski's class. JFM CLASSES2 th. 67 (partly). (Contributed by FL, 15-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( ( T  e.  Tarski  /\  Tr  T ) 
 /\  A  e.  T  /\  B  e.  T ) 
 ->  ( A  ^m  B )  e.  T )
 
Theoremtskurn 8664 A transitive Tarski's class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |-  ( ( ( T  e.  Tarski  /\  Tr  T ) 
 /\  A  e.  T  /\  F : A --> T ) 
 ->  U. ran  F  e.  T )
 
4.1.4  Grothendieck's universes
 
Syntaxcgru 8665 Extend class notation to include the class of all Grothendieck's universes.
 class  Univ
 
Definitiondf-gru 8666* A Grothendieck's universe is a set that is closed with respect to all the operations that are common in set theory: pairs, powersets, unions, intersections, cross 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  /\  A. x  e.  u  ( ~P x  e.  u  /\  A. y  e.  u  { x ,  y }  e.  u  /\  A. y  e.  ( u  ^m  x ) U. ran  y  e.  u ) ) }
 
Theoremelgrug 8667* Properties of a Grothendieck's universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( U  e.  V  ->  ( U  e.  Univ  <->  ( Tr  U  /\  A. x  e.  U  ( ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U  /\  A. y  e.  ( U  ^m  x ) U. ran  y  e.  U ) ) ) )
 
Theoremgrutr 8668 A Grothendieck's universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( U  e.  Univ  ->  Tr  U )
 
Theoremgruelss 8669 A Grothendieck's universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U ) 
 ->  A  C_  U )
 
Theoremgrupw 8670 A Grothendieck's universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U ) 
 ->  ~P A  e.  U )
 
Theoremgruss 8671 Any subset of an element of a Grothendieck's universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  C_  A )  ->  B  e.  U )
 
Theoremgrupr 8672 A Grothendieck's universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A ,  B }  e.  U )
 
Theoremgruurn 8673 A Grothendieck's universe contains the range of any function which takes values in the universe (see gruiun 8674 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U ) 
 ->  U. ran  F  e.  U )
 
Theoremgruiun 8674* If  B (
x ) is a family of elements of  U and the index set  A is an element of  U, then the indexed union  U_ x  e.  A B is also an element of  U, where  U is a Grothendieck's universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  A. x  e.  A  B  e.  U )  ->  U_ x  e.  A  B  e.  U )
 
Theoremgruuni 8675 A Grothendieck's universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U ) 
 ->  U. A  e.  U )
 
Theoremgrurn 8676 A Grothendieck's universe contains the range of any function which takes values in the universe (see gruiun 8674 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U ) 
 ->  ran  F  e.  U )
 
Theoremgruima 8677 A Grothendieck's universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  Fun  F  /\  ( F " A )  C_  U )  ->  ( A  e.  U  ->  ( F " A )  e.  U ) )
 
Theoremgruel 8678 Any element of an element of a Grothendieck's universe is also an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  A )  ->  B  e.  U )
 
Theoremgrusn 8679 A Grothendieck's universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U ) 
 ->  { A }  e.  U )
 
Theoremgruop 8680 A Grothendieck's universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  -> 
 <. A ,  B >.  e.  U )
 
Theoremgruun 8681 A Grothendieck's universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  e.  U )
 
Theoremgruxp 8682 A Grothendieck's universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  X.  B )  e.  U )
 
Theoremgrumap 8683 A Grothendieck's universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  ^m  B )  e.  U )
 
Theoremgruixp 8684* A Grothendieck's universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  A. x  e.  A  B  e.  U )  ->  X_ x  e.  A  B  e.  U )
 
Theoremgruiin 8685* A Grothendieck's universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  E. x  e.  A  B  e.  U )  -> 
 |^|_ x  e.  A  B  e.  U )
 
Theoremgruf 8686 A Grothendieck's universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U ) 
 ->  F  e.  U )
 
Theoremgruen 8687 A Grothendieck's universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  C_  U  /\  ( B  e.  U  /\  B  ~~  A ) )  ->  A  e.  U )
 
Theoremgruwun 8688 A nonempty Grothendieck's universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  e. WUni )
 
Theoremintgru 8689 The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( A  C_  Univ  /\  A  =/=  (/) )  ->  |^| A  e.  Univ )
 
Theoremingru 8690* 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  /\  A. x  e.  A  ( ~P x  e.  A  /\  A. y  e.  A  { x ,  y }  e.  A  /\  A. y
 ( y : x --> A  ->  U. ran  y  e.  A ) ) ) 
 ->  ( U  e.  Univ  ->  ( U  i^i  A )  e.  Univ ) )
 
Theoremwfgru 8691 The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
 |-  ( U  e.  Univ  ->  ( U  i^i  U. ( R1 " On ) )  e.  Univ )
 
Theoremgrudomon 8692 Each ordinal that is comparable with an element of the universe is in the universe. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  On  /\  ( B  e.  U  /\  A  ~<_  B ) ) 
 ->  A  e.  U )
 
Theoremgruina 8693 If a Grothendieck's 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  i^i  On )   =>    |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  Inacc
 )
 
Theoremgrur1a 8694 A characterization of Grothendieck's universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  A  =  ( U  i^i  On )   =>    |-  ( U  e.  Univ 
 ->  ( R1 `  A )  C_  U )
 
Theoremgrur1 8695 A characterization of Grothendieck's universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  =  ( U  i^i  On )   =>    |-  ( ( U  e.  Univ  /\  U  e.  U. ( R1 " On ) )  ->  U  =  ( R1 `  A ) )
 
Theoremgrutsk1 8696 Grothendieck's universes are the same as transitive Tarski's classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 8658.) (Contributed by Mario Carneiro, 17-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\ 
 Tr  T )  ->  T  e.  Univ )
 
Theoremgrutsk 8697 Grothendieck's universes are the same as transitive Tarski's classes. (The proof in the forward direction requires Foundation.) (Contributed by Mario Carneiro, 24-Jun-2013.)
 |- 
 Univ  =  { x  e.  Tarski  |  Tr  x }
 
4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
 
4.2.1  Introduce the Tarski-Grothendieck Axiom
 
Axiomax-groth 8698* 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 8709. An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  ( A. w ( w  C_  z  ->  w  e.  y
 )  /\  E. w  e.  y  A. v ( v  C_  z  ->  v  e.  w ) ) 
 /\  A. z ( z 
 C_  y  ->  (
 z  ~~  y  \/  z  e.  y )
 ) )
 
Theoremaxgroth5 8699* The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
 ( z  ~~  y  \/  z  e.  y
 ) )
 
Theoremaxgroth2 8700* Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  ( A. w ( w  C_  z  ->  w  e.  y
 )  /\  E. w  e.  y  A. v ( v  C_  z  ->  v  e.  w ) ) 
 /\  A. z ( z 
 C_  y  ->  (
 y  ~<_  z  \/  z  e.  y ) ) )
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