HomeHome Metamath Proof Explorer
Theorem List (p. 83 of 313)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21423)
  Hilbert Space Explorer  Hilbert Space Explorer
(21424-22946)
  Users' Mathboxes  Users' Mathboxes
(22947-31284)
 

Theorem List for Metamath Proof Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfpwwe 8201* Given any function  F from the powerset of  A to  A, canth2 6947 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset  <. X , 
( W `  X
) >. which "agrees" with  F in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 7590. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } )
 )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  x  e.  ( ~P A  i^i  dom  card ) )  ->  ( F `  x )  e.  A )   &    |-  X  =  U. dom  W   =>    |-  ( ph  ->  (
 ( Y W R  /\  ( F `  Y )  e.  Y )  <->  ( Y  =  X  /\  R  =  ( W `  X ) ) ) )
 
Theoremcanth4 8202* An "effective" form of Cantor's theorem canth 6225. For any function  F from the powerset of  A to  A, there are two definable sets  B and  C which witness non-injectivity of  F. Corollary 1.3 of [KanamoriPincus] p. 416. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } )
 )  =  y ) ) }   &    |-  B  =  U. dom  W   &    |-  C  =  ( `' ( W `  B ) " { ( F `
  B ) }
 )   =>    |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B )  =  ( F `  C ) ) )
 
Theoremcanthnumlem 8203* Lemma for canthnum 8204. (Contributed by Mario Carneiro, 19-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } )
 )  =  y ) ) }   &    |-  B  =  U. dom  W   &    |-  C  =  ( `' ( W `  B ) " { ( F `
  B ) }
 )   =>    |-  ( A  e.  V  ->  -.  F : ( ~P A  i^i  dom  card
 ) -1-1-> A )
 
Theoremcanthnum 8204 The set of well-orderable subsets of a set  A strictly dominates  A. A stronger form of canth2 6947. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)
 |-  ( A  e.  V  ->  A  ~<  ( ~P A  i^i  dom  card ) )
 
Theoremcanthwelem 8205* Lemma for canthnum 8204. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  O  =  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) }   &    |-  W  =  { <. x ,  r >.  |  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  B  =  U. dom  W   &    |-  C  =  ( `' ( W `  B ) " { ( B F ( W `  B ) ) }
 )   =>    |-  ( A  e.  V  ->  -.  F : O -1-1-> A )
 
Theoremcanthwe 8206* The set of well-orders of a set  A strictly dominates  A. A stronger form of canth2 6947. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  O  =  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) }   =>    |-  ( A  e.  V  ->  A  ~<  O )
 
Theoremcanthp1lem1 8207 Lemma for canthp1 8209. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( 1o  ~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
 
Theoremcanthp1lem2 8208* Lemma for canthp1 8209. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( ph  ->  1o  ~<  A )   &    |-  ( ph  ->  F : ~P A -1-1-onto-> ( A  +c  1o ) )   &    |-  ( ph  ->  G : ( ( A  +c  1o )  \  { ( F `  A ) } ) -1-1-onto-> A )   &    |-  H  =  ( ( G  o.  F )  o.  ( x  e. 
 ~P A  |->  if ( x  =  A ,  (/)
 ,  x ) ) )   &    |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( H `  ( `' r " { y } )
 )  =  y ) ) }   &    |-  B  =  U. dom  W   =>    |- 
 -.  ph
 
Theoremcanthp1 8209 A slightly stronger form of Cantor's theorem: For  1  <  n,  n  +  1  <  2 ^ n. Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( 1o  ~<  A  ->  ( A  +c  1o )  ~<  ~P A )
 
Theoremfinngch 8210 The exclusion of finite sets from consideration in df-gch 8176 is necessary, because otherwise finite sets larger than a singleton would violate the GCH property. (Contributed by Mario Carneiro, 10-Jun-2015.)
 |-  ( ( A  e.  Fin  /\  1o  ~<  A )  ->  ( A  ~<  ( A  +c  1o )  /\  ( A  +c  1o )  ~<  ~P A ) )
 
Theoremgchcda1 8211 An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  ( A  +c  1o )  ~~  A )
 
Theoremgchinf 8212 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  om  ~<_  A )
 
Theorempwfseqlem1 8213* Lemma for pwfseq 8219. Derive a contradiction by diagonalization. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   =>    |-  (
 ( ph  /\  ps )  ->  D  e.  ( U_ n  e.  om  ( A 
 ^m  n )  \  U_ n  e.  om  ( x  ^m  n ) ) )
 
Theorempwfseqlem2 8214* Lemma for pwfseq 8219. (Contributed by Mario Carneiro, 18-Nov-2014.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   &    |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^|
 { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )   =>    |-  ( ( Y  e.  Fin  /\  R  e.  _V )  ->  ( Y F R )  =  ( H `  ( card `  Y )
 ) )
 
Theorempwfseqlem3 8215* Lemma for pwfseq 8219. Using the construction  D from pwfseqlem1 8213, produce a function  F that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   &    |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^|
 { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( x F r )  e.  ( A  \  x ) )
 
Theorempwfseqlem4a 8216* Lemma for pwfseqlem4 8217. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   &    |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^|
 { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )   =>    |-  ( ( ph  /\  (
 a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a ) )  ->  ( a F s )  e.  A )
 
Theorempwfseqlem4 8217* Lemma for pwfseq 8219. Derive a final contradiction from the function  F in pwfseqlem3 8215. Applying fpwwe2 8198 to it, we get a certain maximal well-ordered subset 
Z, but the defining property  ( Z F ( W `  Z
) )  e.  Z contradicts our assumption on  F, so we are reduced to the case of 
Z finite. This too is a contradiction, though, because  Z and its preimage under  ( W `  Z
) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   &    |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^|
 { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )   &    |-  W  =  { <. a ,  s >.  |  (
 ( a  C_  A  /\  s  C_  ( a  X.  a ) ) 
 /\  ( s  We  a  /\  A. b  e.  a  [. ( `' s " { b } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  b ) ) }   &    |-  Z  =  U. dom  W   =>    |- 
 -.  ph
 
Theorempwfseqlem5 8218* Lemma for pwfseq 8219. Although in some ways pwfseqlem4 8217 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection  K from the set of finite sequences on an infinite set 
x to  x. Now this alone would not be difficult to prove; this is mostly the claim of fseqen 7587. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas.

If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 7575. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 7342), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 7578). (Contributed by Mario Carneiro, 31-May-2015.)

 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( t 
 C_  A  /\  r  C_  ( t  X.  t
 )  /\  r  We  t )  /\  om  ~<_  t ) )   &    |-  ( ph  ->  A. b  e.  (har `  ~P A ) ( om  C_  b  ->  ( N `  b ) : ( b  X.  b ) -1-1-onto-> b ) )   &    |-  O  = OrdIso (
 r ,  t )   &    |-  T  =  ( u  e.  dom  O ,  v  e.  dom  O  |->  <. ( O `
  u ) ,  ( O `  v
 ) >. )   &    |-  P  =  ( ( O  o.  ( N `  dom  O ) )  o.  `' T )   &    |-  S  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  ( x  e.  ( t  ^m  suc  k )  |->  ( ( f `  ( x  |`  k ) ) P ( x `  k
 ) ) ) ) ,  { <. (/) ,  ( O `  (/) ) >. } )   &    |-  Q  =  ( y  e.  U_ n  e.  om  ( t 
 ^m  n )  |->  <. dom  y ,  ( ( S `  dom  y
 ) `  y ) >. )   &    |-  I  =  ( x  e.  om ,  y  e.  t  |->  <.
 ( O `  x ) ,  y >. )   &    |-  K  =  ( ( P  o.  I )  o.  Q )   =>    |- 
 -.  ph
 
Theorempwfseq 8219* The powerset of a Dedekind-infinite set does not inject into the set of finite sequences. The proof is due to Halbeisen and Shelah. Proposition 1.7 of [KanamoriPincus] p. 418. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( om  ~<_  A  ->  -. 
 ~P A  ~<_  U_ n  e.  om  ( A  ^m  n ) )
 
Theorempwxpndom2 8220 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( om  ~<_  A  ->  -. 
 ~P A  ~<_  ( A  +c  ( A  X.  A ) ) )
 
Theorempwxpndom 8221 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( om  ~<_  A  ->  -. 
 ~P A  ~<_  ( A  X.  A ) )
 
Theorempwcdandom 8222 The powerset of a Dedekind-infinite set does not inject into its cardinal sum with itself. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( om  ~<_  A  ->  -. 
 ~P A  ~<_  ( A  +c  A ) )
 
Theoremgchcdaidm 8223 An infinite GCH-set is idempotent under cardinal sum. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  ( A  +c  A )  ~~  A )
 
Theoremgchxpidm 8224 An infinite GCH-set is idempotent under cardinal product. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  ( A  X.  A )  ~~  A )
 
Theoremgchaclem 8225 Lemma for gchac 8228 (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 8226 A "local" form of gchac 8228. 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 8227 A "local" form of gchac 8228. 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 8228 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 8229 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 8230 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 8231 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 8232 If  A  +  ~~  ~P A, then  A is a GCH-set. The much simpler converse to gchhar 8226. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  ( (har `  A )  ~~  ~P A  ->  A  e. GCH )
 
Theoremalephgch 8233 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 8234 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 8235 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 8236* 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 8136 to the successor aleph using enen2 6935. (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 8378, 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 8239 and df-ina 8240 respectively), Tarski's classes (df-tsk 8304), and a Grothendieck's universe (df-gru 8346). We then introduce the Tarski's axiom ax-groth 8378 and prove various properties from that.

 
4.1  Inaccessibles
 
4.1.1  Weakly and strongly inaccessible cardinals
 
Syntaxcwina 8237 The class of weak inaccessibles.
 class  Inacc W
 
Syntaxcina 8238 The class of strong inaccessibles.
 class  Inacc
 
Definitiondf-wina 8239* 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 8240* 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 8241* 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 8242* 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 8243 A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  A  e.  On )
 
Theoreminawinalem 8244* Lemma for inawina 8245. (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 8245 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc  ->  A  e.  Inacc W )
 
Theoremomina 8246  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 8247 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  ( card `  A )  =  A )
 
Theoremwinainflem 8248* 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 8249 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  om  C_  A )
 
Theoremwinalim 8250 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  Lim  A )
 
Theoremwinalim2 8251* 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 8252 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 8253 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 3547 to turn this type of statement into the closed form statement winafp 8252, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 8252 using this theorem and dedth 3547, in ZFC. (You can prove this if you use ax-groth 8378, though.) (Contributed by Mario Carneiro, 28-May-2014.)
 |-  A  e.  Inacc W   &    |-  A  =/=  om   =>    |-  ( aleph `  A )  =  A
 
Theoremgchina 8254 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 8255 Extend class definition to include the class of all weak universes.
 class WUni
 
Syntaxcwunm 8256 Extend class definition to include the map whose value is the smallest weak universe.
 class wUniCl
 
Definitiondf-wun 8257* 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 8295 of grothtsk 8390 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 8258* 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 8259* 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 8260 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( U  e. WUni  ->  Tr  U )
 
Theoremwununi 8261 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 8262 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 8263 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 8264 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 8265 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 8266 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 8267 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 8268 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 8269 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 8270 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 8271 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 8272 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 8273 A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   =>    |-  ( ph  ->  (/)  e.  U )
 
Theoremwunr1om 8274 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 8275 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 8276 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 8277 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 8278 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 8279 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 8280 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 8281 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 8282 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 8283 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 8284 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 8285 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 8286 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 8287 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 8288 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 8289 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 8290 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 8291 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 8292 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 8293* 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 8294* 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 8295 Every set is contained in a weak universe. This is the analogue of grothtsk 8390, but it is provable in ZFC without the Tarski-Grothendieck axiom. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 U.WUni  =  _V
 
TheoremwunexALT 8296 Construct a weak universe from a given set. This version of wunex 8294 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 8297* 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 8298 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 8299 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 8300 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 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31284
  Copyright terms: Public domain < Previous  Next >