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Theorem List for Metamath Proof Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwuncidm 8301 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  (wUniCl `  (wUniCl `  A ) )  =  (wUniCl `  A ) )
 
Theoremwuncval2 8302* 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 8303 Extend class definition to include the class of all Tarski's classes.
 class  Tarski
 
Definitiondf-tsk 8304* 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 8378 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 8305* 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 8306* 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 8307 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 8308 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 8309 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 8310 The empty set is a (transitive) Tarski's class. (Contributed by FL, 30-Dec-2010.)
 |-  (/)  e.  Tarski
 
Theoremtsksdom 8311 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 8312 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 8313 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 8314 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 8315 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 8316 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 8317 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 8318 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 8319 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 8320 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 8321 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 8322 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 7272.) (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
 
Theoremtskr1om2 8323 A nonempty Tarski's class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 7272.) (Contributed by NM, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( R1 " om )  C_  T )
 
Theoremtskinf 8324 A nonempty Tarski's class is infinite. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  om 
 ~<_  T )
 
Theoremtskpr 8325 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 8326 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 8327 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 8328 A Tarski's class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
 |-  ( T  e.  Tarski  ->  T  e.  dom  card )
 
Theoreminttsk 8329 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 8330  ( 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 8331 The set of hereditarily finite sets is countable. This is a short proof as a consequence of inar1 8330, which requires AC. See r1om 7803 for a direct proof. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.)
 |-  ( R1 `  om )  ~~  om
 
Theoremrankcf 8332 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 8333  ( 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 8334 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 8335 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 8336 An even more direct relationship than r1tskina 8337 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 8337 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 8338 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 8339 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 8340 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 8341 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 8342 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 8343 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 8344 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 8345 Extend class notation to include the class of all Grothendieck's universes.
 class  Univ
 
Definitiondf-gru 8346* 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 8347* 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 8348 A Grothendieck's universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( U  e.  Univ  ->  Tr  U )
 
Theoremgruelss 8349 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 8350 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 8351 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 8352 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 8353 A Grothendieck's universe contains the range of any function which takes values in the universe (see gruiun 8354 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 8354* 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 8355 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 8356 A Grothendieck's universe contains the range of any function which takes values in the universe (see gruiun 8354 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 8357 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 8358 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 8359 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 8360 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 8361 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 8362 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 8363 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 8364* 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 8365* 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 8366 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 8367 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 8368 A nonempty Grothendieck's universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  e. WUni )
 
Theoremintgru 8369 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 8370* 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 8371 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 8372 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 8373 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 8374 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 8375 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 8376 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 8338.) (Contributed by Mario Carneiro, 17-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\ 
 Tr  T )  ->  T  e.  Univ )
 
Theoremgrutsk 8377 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 Tarksi-Grothendieck Axiom
 
4.2.1  Introduce the Tarksi-Grothendieck Axiom
 
Axiomax-groth 8378* The Tarksi-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 8389. 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 8379* 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 8380* Alternate version of the Tarksi-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 ) ) )
 
4.2.2  Derive the Power Set, Infinity and Choice Axioms
 
Theoremgrothpw 8381* Derive the Axiom of Power Sets ax-pow 4126 from the Tarksi-Grothendieck axiom ax-groth 8378. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4126 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.)
 |- 
 E. y A. z
 ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y )
 
Theoremgrothpwex 8382 Derive the Axiom of Power Sets from the Tarksi-Grothendieck axiom ax-groth 8378. Note that ax-pow 4126 is not used by the proof. Use axpweq 4125 to obtain ax-pow 4126. (Contributed by Gérard Lang, 22-Jun-2009.)
 |- 
 ~P x  e.  _V
 
Theoremaxgroth6 8383* 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.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  ~P z  e.  y ) 
 /\  A. z  e.  ~P  y ( z  ~<  y 
 ->  z  e.  y
 ) )
 
Theoremgrothomex 8384 The Tarksi-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 7277). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.)
 |- 
 om  e.  _V
 
Theoremgrothac 8385 The Tarksi-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 8029). This can be put in a more conventional form via ween 7595 and dfac8 7694. 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.)
 |- 
 dom  card  =  _V
 
Theoremaxgroth3 8386* Alternate version of the Tarksi-Grothendieck Axiom. ax-cc 7994 is used to derive this version. (Contributed by NM, 26-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  \/  z  e.  y ) ) )
 
Theoremaxgroth4 8387* Alternate version of the Tarksi-Grothendieck Axiom. ax-ac 8018 is used to derive this version. (Contributed by NM, 16-Apr-2007.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  E. v  e.  y  A. w ( w  C_  z  ->  w  e.  (
 y  i^i  v )
 )  /\  A. z ( z  C_  y  ->  ( ( y  \  z
 )  ~<_  z  \/  z  e.  y ) ) )
 
Theoremgrothprimlem 8388* Lemma for grothprim 8389. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
 |-  ( { u ,  v }  e.  w  <->  E. g ( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  u  \/  h  =  v )
 ) ) )
 
Theoremgrothprim 8389* The Tarksi-Grothendieck Axiom ax-groth 8378 expanded into set theory primitives using 163 symbols (allowing the defined symbols  /\,  \/,  <->, and  E.). An open problem is whether a shorter equivalent exists (when expanded to primitives). (Contributed by NM, 16-Apr-2007.)
 |- 
 E. y ( x  e.  y  /\  A. z ( ( z  e.  y  ->  E. v
 ( v  e.  y  /\  A. w ( A. u ( u  e.  w  ->  u  e.  z )  ->  ( w  e.  y  /\  w  e.  v ) ) ) )  /\  E. w ( ( w  e.  z  ->  w  e.  y )  ->  ( A. v ( ( v  e.  z  ->  E. t A. u ( E. g
 ( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  v  \/  h  =  u ) ) ) 
 ->  u  =  t
 ) )  /\  (
 v  e.  y  ->  ( v  e.  z  \/  E. u ( u  e.  z  /\  E. g ( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  u  \/  h  =  v )
 ) ) ) ) ) )  \/  z  e.  y ) ) ) )
 
Theoremgrothtsk 8390 The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
 |- 
 U. Tarski  =  _V
 
Theoreminaprc 8391 An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |- 
 Inacc  e/  _V
 
4.2.3  Tarski map function
 
Syntaxctskm 8392 Extend class definition to include the map whose value is the smallest Tarski's class.
 class  tarskiMap
 
Definitiondf-tskm 8393* A function that maps a set  x to the smallest Tarski's class that contains the set. (Contributed by FL, 30-Dec-2010.)
 |-  tarskiMap 
 =  ( x  e. 
 _V  |->  |^| { y  e.  Tarski  |  x  e.  y } )
 
Theoremtskmval 8394* Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
 |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
 
Theoremtskmid 8395 The set  A is an element of the smallest Tarski's class that contains  A. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A ) )
 
Theoremtskmcl 8396 A Tarski's class that contains  A is a Tarski's class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  ( tarskiMap `  A )  e.  Tarski
 
Theoremsstskm 8397* Being a part of  ( tarskiMap `  A
). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( A  e.  V  ->  ( B  C_  ( tarskiMap `  A )  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  C_  x ) ) )
 
Theoremeltskm 8398* Belonging to  ( tarskiMap `  A
). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  ( A  e.  V  ->  ( B  e.  ( tarskiMap `  A )  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )
 ) )
 
PART 5  REAL AND COMPLEX NUMBERS

This section derives the basics of real and complex numbers. We first construct and axiomitize real and complex numbers (e.g., ax-resscn 8727). After that we derive their basic properties, various operations like addition (df-plus 8681) and sine (df-sin 12278), and subsets such as the integers (df-z 9957) and natural numbers (df-n 9680).

 
5.1  Construction and axiomatization of real and complex numbers
 
5.1.1  Dedekind-cut construction of real and complex numbers
 
Syntaxcnpi 8399 The set of positive integers, which is the set of natural numbers  om 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 8699. The actual set of Dedekind cuts is defined by df-np 8538.

 class  N.
 
Syntaxcpli 8400 Positive integer addition.
 class  +N
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