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Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgchaclem 8501 Lemma for gchac 8504 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  om  ~<_  A )   &    |-  ( ph  ->  ~P C  e. GCH )   &    |-  ( ph  ->  ( A  ~<_  C  /\  ( B 
 ~<_  ~P C  ->  ~P A  ~<_  B ) ) )   =>    |-  ( ph  ->  ( A  ~<_  ~P C  /\  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) ) )
 
Theoremgchhar 8502 A "local" form of gchac 8504. If  A and  ~P A are GCH-sets, then the Hartogs number of  A is  ~P A (so  ~P A and a fortiori 
A are well-orderable). The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A )  ~~  ~P A )
 
Theoremgchacg 8503 A "local" form of gchac 8504. If  A and  ~P A are GCH-sets, then  ~P A is well-orderable. The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  e.  dom  card )
 
Theoremgchac 8504 The Generalized Continuum Hypothesis implies the Axiom of Choice. The original proof is due to Sierpiński (1947); we use a refinement of Sierpiński's result due to Specker. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  (GCH  =  _V  -> CHOICE )
 
Theoremgchpwdom 8505 A relationship between dominance over the powerset and strict dominance when the sets involved are infinite GCH-sets. Proposition 3.1 of [KanamoriPincus] p. 421. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH ) 
 ->  ( A  ~<  B  <->  ~P A  ~<_  B ) )
 
Theoremgchaleph 8506 If  ( aleph `  A
) is a GCH-set and its powerset is well-orderable, then the successor aleph  ( aleph `  suc  A ) is equinumerous to the powerset of  ( aleph `  A
). (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card
 )  ->  ( aleph ` 
 suc  A )  ~~  ~P ( aleph `  A )
 )
 
Theoremgchaleph2 8507 If  ( aleph `  A
) and  ( aleph `  suc  A ) are GCH-sets, then the successor aleph  ( aleph `  suc  A ) is equinumerous to the powerset of  ( aleph `  A
). (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph ` 
 suc  A )  e. GCH )  ->  ( aleph `  suc  A ) 
 ~~  ~P ( aleph `  A ) )
 
Theoremhargch 8508 If  A  +  ~~  ~P A, then  A is a GCH-set. The much simpler converse to gchhar 8502. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  ( (har `  A )  ~~  ~P A  ->  A  e. GCH )
 
Theoremalephgch 8509 If  ( aleph `  suc  A ) is equinumerous to the powerset of  ( aleph `  A
), then  ( aleph `  A
) is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( aleph `  suc  A )  ~~  ~P ( aleph `  A )  ->  ( aleph `  A )  e. GCH )
 
Theoremgch2 8510 It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.)
 |-  (GCH  =  _V  <->  ran  aleph  C_ GCH )
 
Theoremgch3 8511 An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  (GCH  =  _V  <->  A. x  e.  On  ( aleph `  suc  x ) 
 ~~  ~P ( aleph `  x ) )
 
Theoremgch-kn 8512* The equivalence of two versions of the Generalized Continuum Hypothesis. The right-hand side is the standard version in the literature. The left-hand side is a version devised by Kannan Nambiar, which he calls the Axiom of Combinatorial Sets. For the notation and motivation behind this axiom, see his paper, "Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets," available at http://www.e-atheneum.net/science/derivation_ch.pdf. The equivalence of the two sides provides a negative answer to Open Problem 2 in http://www.e-atheneum.net/science/open_problem_print.pdf. The key idea in the proof below is to equate both sides of alephexp2 8412 to the successor aleph using enen2 7207. (Contributed by NM, 1-Oct-2004.)
 |-  ( A  e.  On  ->  ( ( aleph `  suc  A )  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A )
 ) }  <->  ( aleph `  suc  A )  ~~  ( 2o 
 ^m  ( aleph `  A ) ) ) )
 
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY

Here we introduce Tarski-Grothendieck (TG) set theory, named after mathematicians Alfred Tarski and Alexander Grothendieck. TG theory extends ZFC with the TG Axiom ax-groth 8654, which states that 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."

We first introduce the concept of inaccessibles, including Weakly and strongly inaccessible cardinals (df-wina 8515 and df-ina 8516 respectively), Tarski's classes (df-tsk 8580), and a Grothendieck's universe (df-gru 8622). We then introduce the Tarski's axiom ax-groth 8654 and prove various properties from that.

 
4.1  Inaccessibles
 
4.1.1  Weakly and strongly inaccessible cardinals
 
Syntaxcwina 8513 The class of weak inaccessibles.
 class  Inacc W
 
Syntaxcina 8514 The class of strong inaccessibles.
 class  Inacc
 
Definitiondf-wina 8515* An ordinal is weakly inaccessible iff it is a regular limit cardinal. Note that our definition allows  om as a weakly inacessible cardinal. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |- 
 Inacc W  =  { x  |  ( x  =/= 
 (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z ) }
 
Definitiondf-ina 8516* An ordinal is strongly inaccessible iff it is a regular strong limit cardinal, which is to say that it dominates the powersets of every smaller ordinal. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |- 
 Inacc  =  { x  |  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x ) }
 
Theoremelwina 8517* Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |-  ( A  e.  Inacc W  <-> 
 ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
 
Theoremelina 8518* Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |-  ( A  e.  Inacc  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
 
Theoremwinaon 8519 A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  A  e.  On )
 
Theoreminawinalem 8520* Lemma for inawina 8521. (Contributed by Mario Carneiro, 8-Jun-2014.)
 |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A 
 ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
 
Theoreminawina 8521 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc  ->  A  e.  Inacc W )
 
Theoremomina 8522  om is a strongly inaccessible cardinal. (Many definitions of "inaccessible" explicitly disallow  om as an inaccessible cardinal, but this choice allows us to reuse our results for inaccessibles for  om.) (Contributed by Mario Carneiro, 29-May-2014.)
 |- 
 om  e.  Inacc
 
Theoremwinacard 8523 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  ( card `  A )  =  A )
 
Theoremwinainflem 8524* A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( ( A  =/=  (/)  /\  A  e.  On  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  om  C_  A )
 
Theoremwinainf 8525 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  om  C_  A )
 
Theoremwinalim 8526 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  Lim  A )
 
Theoremwinalim2 8527* A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( ( A  e.  Inacc W  /\  A  =/=  om )  ->  E. x ( (
 aleph `  x )  =  A  /\  Lim  x ) )
 
Theoremwinafp 8528 A nontrivial weakly inaccessible cardinal is a fixed point of the aleph function. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( ( A  e.  Inacc W  /\  A  =/=  om )  ->  ( aleph `  A )  =  A )
 
Theoremwinafpi 8529 This theorem, which states that a nontrivial inaccessible cardinal is its own aleph number, is stated here in inference form, where the assumptions are in the hypotheses rather than an antecedent. Often, we use dedth 3740 to turn this type of statement into the closed form statement winafp 8528, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 8528 using this theorem and dedth 3740, in ZFC. (You can prove this if you use ax-groth 8654, though.) (Contributed by Mario Carneiro, 28-May-2014.)
 |-  A  e.  Inacc W   &    |-  A  =/=  om   =>    |-  ( aleph `  A )  =  A
 
Theoremgchina 8530 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 8531 Extend class definition to include the class of all weak universes.
 class WUni
 
Syntaxcwunm 8532 Extend class definition to include the map whose value is the smallest weak universe.
 class wUniCl
 
Definitiondf-wun 8533* 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 8571 of grothtsk 8666 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 8534* 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 8535* 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 8536 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( U  e. WUni  ->  Tr  U )
 
Theoremwununi 8537 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 8538 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 8539 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 8540 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 8541 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 8542 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 8543 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 8544 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 8545 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 8546 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 8547 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 8548 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 8549 A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   =>    |-  ( ph  ->  (/)  e.  U )
 
Theoremwunr1om 8550 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 8551 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 8552 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 8553 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 8554 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 8555 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 8556 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 8557 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 8558 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 8559 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 8560 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 8561 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 8562 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 8563 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 8564 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 8565 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 8566 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 8567 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 8568 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 8569* 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 8570* Construct a weak universe from a given set. See also wunex2 8569. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  E. u  e. WUni  A  C_  u )
 
Theoremuniwun 8571 Every set is contained in a weak universe. This is the analogue of grothtsk 8666, but it is provable in ZFC without the Tarski-Grothendieck axiom. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 U.WUni  =  _V
 
Theoremwunex3 8572 Construct a weak universe from a given set. This version of wunex 8570 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 8573* 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 8574 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 8575 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 8576 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 8577 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  V  ->  (wUniCl `  (wUniCl `  A ) )  =  (wUniCl `  A ) )
 
Theoremwuncval2 8578* 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 8579 Extend class definition to include the class of all Tarski's classes.
 class  Tarski
 
Definitiondf-tsk 8580* 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 8654 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 8581* 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 8582* 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 8583 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 8584 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 8585 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 8586 The empty set is a (transitive) Tarski's class. (Contributed by FL, 30-Dec-2010.)
 |-  (/)  e.  Tarski
 
Theoremtsksdom 8587 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 8588 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 8589 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 8590 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 8591 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 8592 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 8593 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 8594 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 8595 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 8596 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 8597 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 8598 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 7549.) (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  ( R1 " om )  C_  T )
 
Theoremtskr1om2 8599 A nonempty Tarski's class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 7549.) (Contributed by NM, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  U. ( R1 " om )  C_  T )
 
Theoremtskinf 8600 A nonempty Tarski's class is infinite. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  om 
 ~<_  T )
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