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Theorem List for Metamath Proof Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgchina 8201 Assuming the GCH, weakly and strongly inaccessible cardinals coincide. Theorem 11.20 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 5-Jun-2015.)
 |-  (GCH  =  _V  ->  Inacc W  =  Inacc )
 
4.1.2  Weak universes
 
Syntaxcwun 8202 Extend class definition to include the class of all weak universes.
 class WUni
 
Syntaxcwunm 8203 Extend class definition to include the map whose value is the smallest weak universe.
 class wUniCl
 
Definitiondf-wun 8204* The class of all weak universes. A weak universe is a nonempty transitive class closed under union, pairing, and powerset. The advantage of a weak universe over a Grothendieck universe is that weak universes satisfy the analogue uniwun 8242 of grothtsk 8337 in ZFC. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- WUni  =  { u  |  ( Tr  u  /\  u  =/= 
 (/)  /\  A. x  e.  u  ( U. x  e.  u  /\  ~P x  e.  u  /\  A. y  e.  u  { x ,  y }  e.  u ) ) }
 
Definitiondf-wunc 8205* A function that maps a set  x to the smallest weak universe that contains the elements of the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- wUniCl  =  ( x  e.  _V  |->  |^|
 { u  e. WUni  |  x  C_  u } )
 
Theoremiswun 8206* Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( U  e.  V  ->  ( U  e. WUni  <->  ( Tr  U  /\  U  =/=  (/)  /\  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) ) )
 
Theoremwuntr 8207 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( U  e. WUni  ->  Tr  U )
 
Theoremwununi 8208 A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  U. A  e.  U )
 
Theoremwunpw 8209 A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ~P A  e.  U )
 
Theoremwunelss 8210 The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  A 
 C_  U )
 
Theoremwunpr 8211 A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   =>    |-  ( ph  ->  { A ,  B }  e.  U )
 
Theoremwunun 8212 A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   =>    |-  ( ph  ->  ( A  u.  B )  e.  U )
 
Theoremwuntp 8213 A weak universe is closed under unordered triple. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  { A ,  B ,  C }  e.  U )
 
Theoremwunss 8214 A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  C_  A )   =>    |-  ( ph  ->  B  e.  U )
 
Theoremwunin 8215 A weak universe is closed under intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ( A  i^i  B )  e.  U )
 
Theoremwundif 8216 A weak universe is closed under set difference. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ( A  \  B )  e.  U )
 
Theoremwunint 8217 A weak universe is closed under intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ( ph  /\  A  =/=  (/) )  ->  |^| A  e.  U )
 
Theoremwunsn 8218 A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  { A }  e.  U )
 
Theoremwunsuc 8219 A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  suc 
 A  e.  U )
 
Theoremwun0 8220 A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   =>    |-  ( ph  ->  (/)  e.  U )
 
Theoremwunr1om 8221 A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   =>    |-  ( ph  ->  ( R1 " om )  C_  U )
 
Theoremwunom 8222 A weak universe contains all the finite ordinals, and hence is infinite. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   =>    |-  ( ph  ->  om  C_  U )
 
Theoremwunfi 8223 A weak universe contains all finite sets with elements drawn from the universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  A  e.  U )
 
Theoremwunop 8224 A weak universe is closed under ordered pairs. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   =>    |-  ( ph  ->  <. A ,  B >.  e.  U )
 
Theoremwunot 8225 A weak universe is closed under ordered triples. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  <. A ,  B ,  C >.  e.  U )
 
Theoremwunxp 8226 A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   =>    |-  ( ph  ->  ( A  X.  B )  e.  U )
 
Theoremwunpm 8227 A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   =>    |-  ( ph  ->  ( A  ^pm  B )  e.  U )
 
Theoremwunmap 8228 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 8229 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 8230 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 8231 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 8232 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 8233 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 8234 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 8235 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 8236 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 8237 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 8238 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 8239 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 8240* 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 8241* Construct a weak universe from a given set. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  E. u  e. WUni  A  C_  u )
 
Theoremuniwun 8242 Every set is contained in a weak universe. This is the analogue of grothtsk 8337, but it is provable in ZFC without the Tarski-Grothendieck axiom. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 U.WUni  =  _V
 
TheoremwunexALT 8243 Construct a weak universe from a given set. This version of wunex 8241 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 8244* 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 8245 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 8246 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 8247 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 8248 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  (wUniCl `  (wUniCl `  A ) )  =  (wUniCl `  A ) )
 
Theoremwuncval2 8249* 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 8250 Extend class definition to include the class of all Tarski's classes.
 class  Tarski
 
Definitiondf-tsk 8251* 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 8325 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 8252* 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 8253* 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 8254 1st axiom of a 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 8255 2nd axiom of a 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 8256 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 8257 The empty set is a (transitive) Tarski's class. (Contributed by FL, 30-Dec-2010.)
 |-  (/)  e.  Tarski
 
Theoremtsksdom 8258 A 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 8259 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 8260 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 8261 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 8262 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 8263 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 8264 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 8265 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 8266 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 8267 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 8268 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 8269 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 7223.) (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
 
Theoremtskr1om2 8270 A nonempty Tarski's class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 7223.) (Contributed by NM, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( R1 " om )  C_  T )
 
Theoremtskinf 8271 A nonempty Tarski's class is infinite. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  om 
 ~<_  T )
 
Theoremtskpr 8272 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 8273 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 8274 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 8275 A Tarski's class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
 |-  ( T  e.  Tarski  ->  T  e.  dom  card )
 
Theoreminttsk 8276 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 8277  ( 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 8278 The set of hereditarily finite sets is countable. This is a short proof as a consequence of inar1 8277, which requires AC. See r1om 7754 for a direct proof. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.)
 |-  ( R1 `  om )  ~~  om
 
Theoremrankcf 8279 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 8280  ( 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 8281 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 8282 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 8283 An even more direct relationship than r1tskina 8284 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 8284 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 8285 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 8286 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 8287 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 8288 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 8289 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 8290 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 8291 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 8292 Extend class notation to include the class of all Grothendieck's universes.
 class  Univ
 
Definitiondf-gru 8293* 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 8294* 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 8295 A Grothendieck's universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( U  e.  Univ  ->  Tr  U )
 
Theoremgruelss 8296 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 8297 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 8298 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 8299 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 8300 A Grothendieck's universe contains the range of any function which takes values in the universe (see gruiun 8301 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 )
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