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Theorem List for Metamath Proof Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremenfin2i 7901 II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( A  e. FinII  ->  B  e. FinII ) )
 
Theoremfin23lem24 7902 Lemma for fin23 7969. In a class of ordinals, each element is fully identified by those of its predecessors which also belong to the class. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  ( ( ( Ord 
 A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  ->  ( ( C  i^i  B )  =  ( D  i^i  B )  <->  C  =  D ) )
 
Theoremfincssdom 7903 In a chain of finite sets, dominance and subset coincide. (Contributed by Stefan O'Rear, 8-Nov-2014.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) ) 
 ->  ( A  ~<_  B  <->  A  C_  B ) )
 
Theoremfin23lem25 7904 Lemma for fin23 7969. In a chain of finite sets, equinumerousity is equivalent to equality. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) ) 
 ->  ( A  ~~  B  <->  A  =  B ) )
 
Theoremfin23lem26 7905* Lemma for fin23lem22 7907. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  ( ( ( S 
 C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E. j  e.  S  ( j  i^i 
 S )  ~~  i
 )
 
Theoremfin23lem23 7906* Lemma for fin23lem22 7907. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  ( ( ( S 
 C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E! j  e.  S  (
 j  i^i  S )  ~~  i )
 
Theoremfin23lem22 7907* Lemma for fin23 7969 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 7908) between an infinite subset of  om and  om itself. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S ( j  i^i  S )  ~~  i ) )   =>    |-  ( ( S 
 C_  om  /\  -.  S  e.  Fin )  ->  C : om
 -1-1-onto-> S )
 
Theoremfin23lem27 7908* The mapping constructed in fin23lem22 7907 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S ( j  i^i  S )  ~~  i ) )   =>    |-  ( ( S 
 C_  om  /\  -.  S  e.  Fin )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
 
Theoremisfin3ds 7909* Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. b  e.  om  ( a `  suc  b )  C_  ( a `
  b )  ->  |^| ran  a  e.  ran  a ) }   =>    |-  ( A  e.  V  ->  ( A  e.  F 
 <-> 
 A. f  e.  ( ~P A  ^m  om )
 ( A. x  e.  om  ( f `  suc  x )  C_  ( f `  x )  ->  |^| ran  f  e.  ran  f ) ) )
 
Theoremssfin3ds 7910* A subset of a III-finite set is III-finite. (Contributed by Stefan O'Rear, 4-Nov-2014.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. b  e.  om  ( a `  suc  b )  C_  ( a `
  b )  ->  |^| ran  a  e.  ran  a ) }   =>    |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  F )
 
Theoremfin23lem12 7911* The beginning of the proof that every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets).

This first section is dedicated to the construction of  U and its intersection. First, the value of  U at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)

 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   =>    |-  ( A  e.  om  ->  ( U `  suc  A )  =  if (
 ( ( t `  A )  i^i  ( U `
  A ) )  =  (/) ,  ( U `
  A ) ,  ( ( t `  A )  i^i  ( U `
  A ) ) ) )
 
Theoremfin23lem13 7912* Lemma for fin23 7969. Each step of  U is a decrease. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   =>    |-  ( A  e.  om  ->  ( U `  suc  A )  C_  ( U `  A ) )
 
Theoremfin23lem14 7913* Lemma for fin23 7969. 
U will never evolve to an empty set if it did not start with one. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   =>    |-  ( ( A  e.  om 
 /\  U. ran  t  =/=  (/) )  ->  ( U `
  A )  =/=  (/) )
 
Theoremfin23lem15 7914* Lemma for fin23 7969. 
U is a monotone function. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   =>    |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( U `  A )  C_  ( U `  B ) )
 
Theoremfin23lem16 7915* Lemma for fin23 7969. 
U ranges over the original set; in particular  ran  U is a set, although we do not assume here that  U is. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   =>    |- 
 U. ran  U  =  U.
 ran  t
 
Theoremfin23lem19 7916* Lemma for fin23 7969. The first set in  U to see an input set is either contained in it or disjoint from it. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   =>    |-  ( A  e.  om  ->  ( ( U `  suc  A )  C_  (
 t `  A )  \/  ( ( U `  suc  A )  i^i  (
 t `  A )
 )  =  (/) ) )
 
Theoremfin23lem20 7917* Lemma for fin23 7969. 
X is either contained in or disjoint from all input sets. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   =>    |-  ( A  e.  om  ->  ( |^| ran  U  C_  ( t `  A )  \/  ( |^| ran  U  i^i  ( t `  A ) )  =  (/) ) )
 
Theoremfin23lem17 7918* Lemma for fin23 7969. By ? Fin3DS ? ,  U achieves its minimum ( X in the synopsis above, but we will not be assigning a symbol here). TODO: Fix comment; math symbol Fin3DS does not exist. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   &    |-  F  =  {
 g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
 a `  x )  -> 
 |^| ran  a  e.  ran  a ) }   =>    |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V ) 
 ->  |^| ran  U  e.  ran 
 U )
 
Theoremfin23lem21 7919* Lemma for fin23 7969. 
X is not empty. We only need here that  t has at least one set in its range besides  (/); the much stronger hypothesis here will serve as our induction hypothesis though. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   &    |-  F  =  {
 g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
 a `  x )  -> 
 |^| ran  a  e.  ran  a ) }   =>    |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V ) 
 ->  |^| ran  U  =/=  (/) )
 
Theoremfin23lem28 7920* Lemma for fin23 7969. The residual is also one-to-one. This preserves the induction invariant. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   &    |-  F  =  {
 g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
 a `  x )  -> 
 |^| ran  a  e.  ran  a ) }   &    |-  P  =  { v  e.  om  |  |^| ran  U  C_  (
 t `  v ) }   &    |-  Q  =  ( w  e.  om  |->  ( iota_ x  e.  P ( x  i^i  P )  ~~  w ) )   &    |-  R  =  ( w  e.  om  |->  ( iota_ x  e.  ( om  \  P ) ( x  i^i  ( om  \  P ) )  ~~  w ) )   &    |-  Z  =  if ( P  e.  Fin
 ,  ( t  o.  R ) ,  (
 ( z  e.  P  |->  ( ( t `  z )  \  |^| ran  U ) )  o.  Q ) )   =>    |-  ( t : om -1-1-> _V 
 ->  Z : om -1-1-> _V )
 
Theoremfin23lem29 7921* Lemma for fin23 7969. The residual is built from the same elements as the previous sequence. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   &    |-  F  =  {
 g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
 a `  x )  -> 
 |^| ran  a  e.  ran  a ) }   &    |-  P  =  { v  e.  om  |  |^| ran  U  C_  (
 t `  v ) }   &    |-  Q  =  ( w  e.  om  |->  ( iota_ x  e.  P ( x  i^i  P )  ~~  w ) )   &    |-  R  =  ( w  e.  om  |->  ( iota_ x  e.  ( om  \  P ) ( x  i^i  ( om  \  P ) )  ~~  w ) )   &    |-  Z  =  if ( P  e.  Fin
 ,  ( t  o.  R ) ,  (
 ( z  e.  P  |->  ( ( t `  z )  \  |^| ran  U ) )  o.  Q ) )   =>    |- 
 U. ran  Z  C_  U. ran  t
 
Theoremfin23lem30 7922* Lemma for fin23 7969. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   &    |-  F  =  {
 g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
 a `  x )  -> 
 |^| ran  a  e.  ran  a ) }   &    |-  P  =  { v  e.  om  |  |^| ran  U  C_  (
 t `  v ) }   &    |-  Q  =  ( w  e.  om  |->  ( iota_ x  e.  P ( x  i^i  P )  ~~  w ) )   &    |-  R  =  ( w  e.  om  |->  ( iota_ x  e.  ( om  \  P ) ( x  i^i  ( om  \  P ) )  ~~  w ) )   &    |-  Z  =  if ( P  e.  Fin
 ,  ( t  o.  R ) ,  (
 ( z  e.  P  |->  ( ( t `  z )  \  |^| ran  U ) )  o.  Q ) )   =>    |-  ( Fun  t  ->  ( U. ran  Z  i^i  |^|
 ran  U )  =  (/) )
 
Theoremfin23lem31 7923* Lemma for fin23 7969. The residual is has a strictly smaller range than the previous sequence. This will be iterated to build an unbounded chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   &    |-  F  =  {
 g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
 a `  x )  -> 
 |^| ran  a  e.  ran  a ) }   &    |-  P  =  { v  e.  om  |  |^| ran  U  C_  (
 t `  v ) }   &    |-  Q  =  ( w  e.  om  |->  ( iota_ x  e.  P ( x  i^i  P )  ~~  w ) )   &    |-  R  =  ( w  e.  om  |->  ( iota_ x  e.  ( om  \  P ) ( x  i^i  ( om  \  P ) )  ~~  w ) )   &    |-  Z  =  if ( P  e.  Fin
 ,  ( t  o.  R ) ,  (
 ( z  e.  P  |->  ( ( t `  z )  \  |^| ran  U ) )  o.  Q ) )   =>    |-  ( ( t : om -1-1-> V  /\  G  e.  F  /\  U. ran  t  C_  G )  ->  U. ran  Z 
 C.  U. ran  t )
 
Theoremfin23lem32 7924* Lemma for fin23 7969. Wrap the previous construction into a function to hide the hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  U  = seq𝜔 ( ( i  e. 
 om ,  u  e. 
 _V  |->  if ( ( ( t `  i )  i^i  u )  =  (/) ,  u ,  (
 ( t `  i
 )  i^i  u )
 ) ) ,  U. ran  t )   &    |-  F  =  {
 g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
 a `  x )  -> 
 |^| ran  a  e.  ran  a ) }   &    |-  P  =  { v  e.  om  |  |^| ran  U  C_  (
 t `  v ) }   &    |-  Q  =  ( w  e.  om  |->  ( iota_ x  e.  P ( x  i^i  P )  ~~  w ) )   &    |-  R  =  ( w  e.  om  |->  ( iota_ x  e.  ( om  \  P ) ( x  i^i  ( om  \  P ) )  ~~  w ) )   &    |-  Z  =  if ( P  e.  Fin
 ,  ( t  o.  R ) ,  (
 ( z  e.  P  |->  ( ( t `  z )  \  |^| ran  U ) )  o.  Q ) )   =>    |-  ( G  e.  F  ->  E. f A. b
 ( ( b : om -1-1-> _V  /\  U. ran  b  C_  G )  ->  ( ( f `  b ) : om -1-1-> _V 
 /\  U. ran  (  f `
  b )  C.  U.
 ran  b ) ) )
 
Theoremfin23lem33 7925* Lemma for fin23 7969. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   =>    |-  ( G  e.  F  ->  E. f A. b
 ( ( b : om -1-1-> _V  /\  U. ran  b  C_  G )  ->  ( ( f `  b ) : om -1-1-> _V 
 /\  U. ran  (  f `
  b )  C.  U.
 ran  b ) ) )
 
Theoremfin23lem34 7926* Lemma for fin23 7969. Establish induction invariants on  Y which parameterizes our contradictory chain of subsets. In this section,  h is the hypothetically assumed family of subsets,  g is the ground set, and  i is the induction function constructed in the previous section. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   &    |-  ( ph  ->  h : om -1-1-> _V )   &    |-  ( ph  ->  U. ran  h  C_  G )   &    |-  ( ph  ->  A. j ( ( j : om -1-1-> _V  /\  U.
 ran  j  C_  G )  ->  ( ( i `
  j ) : om -1-1-> _V  /\  U. ran  (  i `  j ) 
 C.  U. ran  j ) ) )   &    |-  Y  =  ( rec ( i ,  h )  |`  om )   =>    |-  (
 ( ph  /\  A  e.  om )  ->  ( ( Y `  A ) : om -1-1-> _V  /\  U. ran  (  Y `  A ) 
 C_  G ) )
 
Theoremfin23lem35 7927* Lemma for fin23 7969. Strict order property of  Y. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   &    |-  ( ph  ->  h : om -1-1-> _V )   &    |-  ( ph  ->  U. ran  h  C_  G )   &    |-  ( ph  ->  A. j ( ( j : om -1-1-> _V  /\  U.
 ran  j  C_  G )  ->  ( ( i `
  j ) : om -1-1-> _V  /\  U. ran  (  i `  j ) 
 C.  U. ran  j ) ) )   &    |-  Y  =  ( rec ( i ,  h )  |`  om )   =>    |-  (
 ( ph  /\  A  e.  om )  ->  U. ran  (  Y `  suc  A ) 
 C.  U. ran  (  Y `
  A ) )
 
Theoremfin23lem36 7928* Lemma for fin23 7969. Weak order property of  Y. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   &    |-  ( ph  ->  h : om -1-1-> _V )   &    |-  ( ph  ->  U. ran  h  C_  G )   &    |-  ( ph  ->  A. j ( ( j : om -1-1-> _V  /\  U.
 ran  j  C_  G )  ->  ( ( i `
  j ) : om -1-1-> _V  /\  U. ran  (  i `  j ) 
 C.  U. ran  j ) ) )   &    |-  Y  =  ( rec ( i ,  h )  |`  om )   =>    |-  (
 ( ( A  e.  om 
 /\  B  e.  om )  /\  ( B  C_  A  /\  ph ) )  ->  U. ran  (  Y `  A )  C_  U. ran  (  Y `  B ) )
 
Theoremfin23lem38 7929* Lemma for fin23 7969. The contradictory chain has no minimum. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   &    |-  ( ph  ->  h : om -1-1-> _V )   &    |-  ( ph  ->  U. ran  h  C_  G )   &    |-  ( ph  ->  A. j ( ( j : om -1-1-> _V  /\  U.
 ran  j  C_  G )  ->  ( ( i `
  j ) : om -1-1-> _V  /\  U. ran  (  i `  j ) 
 C.  U. ran  j ) ) )   &    |-  Y  =  ( rec ( i ,  h )  |`  om )   =>    |-  ( ph  ->  -.  |^| ran  (  b  e.  om  |->  U. ran  (  Y `  b ) )  e.  ran  (  b  e.  om  |->  U. ran  (  Y `  b ) ) )
 
Theoremfin23lem39 7930* Lemma for fin23 7969. Thus we have that  g could not have been in  F after all. (Contributed by Stefan O'Rear, 4-Nov-2014.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   &    |-  ( ph  ->  h : om -1-1-> _V )   &    |-  ( ph  ->  U. ran  h  C_  G )   &    |-  ( ph  ->  A. j ( ( j : om -1-1-> _V  /\  U.
 ran  j  C_  G )  ->  ( ( i `
  j ) : om -1-1-> _V  /\  U. ran  (  i `  j ) 
 C.  U. ran  j ) ) )   &    |-  Y  =  ( rec ( i ,  h )  |`  om )   =>    |-  ( ph  ->  -.  G  e.  F )
 
Theoremfin23lem40 7931* Lemma for fin23 7969. FinII sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   =>    |-  ( A  e. FinII  ->  A  e.  F )
 
Theoremfin23lem41 7932* Lemma for fin23 7969. A set which satisfies the descending sequence condition must be III-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   =>    |-  ( A  e.  F  ->  A  e. FinIII )
 
Theoremisf32lem1 7933* Lemma for isfin3-2 7947. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   =>    |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  C_  A  /\  ph ) )  ->  ( F `  A ) 
 C_  ( F `  B ) )
 
Theoremisf32lem2 7934* Lemma for isfin3-2 7947. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   =>    |-  ( ( ph  /\  A  e.  om )  ->  E. a  e.  om  ( A  e.  a  /\  ( F `  suc  a )  C.  ( F `
  a ) ) )
 
Theoremisf32lem3 7935* Lemma for isfin3-2 7947. Being a chain, difference sets are disjoint (one case). (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   =>    |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  ( ( ( F `
  A )  \  ( F `  suc  A ) )  i^i  ( ( F `  B ) 
 \  ( F `  suc  B ) ) )  =  (/) )
 
Theoremisf32lem4 7936* Lemma for isfin3-2 7947. Being a chain, difference sets are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   =>    |-  ( ( (
 ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  ->  ( ( ( F `
  A )  \  ( F `  suc  A ) )  i^i  ( ( F `  B ) 
 \  ( F `  suc  B ) ) )  =  (/) )
 
Theoremisf32lem5 7937* Lemma for isfin3-2 7947. There are infinite decrease points. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   =>    |-  ( ph  ->  -.  S  e.  Fin )
 
Theoremisf32lem6 7938* Lemma for isfin3-2 7947. Each K value is non-empty. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   &    |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S ( v  i^i  S ) 
 ~~  u ) )   &    |-  K  =  ( ( w  e.  S  |->  ( ( F `  w ) 
 \  ( F `  suc  w ) ) )  o.  J )   =>    |-  ( ( ph  /\  A  e.  om )  ->  ( K `  A )  =/=  (/) )
 
Theoremisf32lem7 7939* Lemma for isfin3-2 7947. Different K values are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   &    |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S ( v  i^i  S ) 
 ~~  u ) )   &    |-  K  =  ( ( w  e.  S  |->  ( ( F `  w ) 
 \  ( F `  suc  w ) ) )  o.  J )   =>    |-  ( ( (
 ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  ->  ( ( K `  A )  i^i  ( K `
  B ) )  =  (/) )
 
Theoremisf32lem8 7940* Lemma for isfin3-2 7947. K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   &    |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S ( v  i^i  S ) 
 ~~  u ) )   &    |-  K  =  ( ( w  e.  S  |->  ( ( F `  w ) 
 \  ( F `  suc  w ) ) )  o.  J )   =>    |-  ( ( ph  /\  A  e.  om )  ->  ( K `  A )  C_  G )
 
Theoremisf32lem9 7941* Lemma for isfin3-2 7947. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   &    |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S ( v  i^i  S ) 
 ~~  u ) )   &    |-  K  =  ( ( w  e.  S  |->  ( ( F `  w ) 
 \  ( F `  suc  w ) ) )  o.  J )   &    |-  L  =  ( t  e.  G  |->  ( iota s ( s  e.  om  /\  t  e.  ( K `  s
 ) ) ) )   =>    |-  ( ph  ->  L : G -onto-> om )
 
Theoremisf32lem10 7942* Lemma for isfin3-2 . Write in terms of weak dominance. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   &    |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S ( v  i^i  S ) 
 ~~  u ) )   &    |-  K  =  ( ( w  e.  S  |->  ( ( F `  w ) 
 \  ( F `  suc  w ) ) )  o.  J )   &    |-  L  =  ( t  e.  G  |->  ( iota s ( s  e.  om  /\  t  e.  ( K `  s
 ) ) ) )   =>    |-  ( ph  ->  ( G  e.  V  ->  om  ~<_*  G ) )
 
Theoremisf32lem11 7943* Lemma for isfin3-2 7947. Remove hypotheses from isf32lem10 7942. (Contributed by Stefan O'Rear, 17-May-2015.)
 |-  ( ( G  e.  V  /\  ( F : om
 --> ~P G  /\  A. b  e.  om  ( F `
  suc  b )  C_  ( F `  b
 )  /\  -.  |^| ran  F  e.  ran  F )
 )  ->  om  ~<_*  G )
 
Theoremisf32lem12 7944* Lemma for isfin3-2 7947. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   =>    |-  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  G  e.  F ) )
 
Theoremisfin32i 7945 One half of isfin3-2 7947. (Contributed by Mario Carneiro, 3-Jun-2015.)
 |-  ( A  e. FinIII  ->  -.  om  ~<_*  A )
 
Theoremisf33lem 7946* Lemma for isfin3-3 7948. (Contributed by Stefan O'Rear, 17-May-2015.)
 |- FinIII  =  { g  |  A. a  e.  ( ~P g  ^m  om ) (
 A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }
 
Theoremisfin3-2 7947 Weakly Dedekind-infinite sets are exactly those which can be mapped onto  om. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinIII  <->  -.  om  ~<_*  A ) )
 
Theoremisfin3-3 7948* Weakly Dedekind-infinite sets are exactly those with an  om-indexed descending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  ( A  e.  V  ->  ( A  e. FinIII  <->  A. f  e.  ( ~P A  ^m  om )
 ( A. x  e.  om  ( f `  suc  x )  C_  ( f `  x )  ->  |^| ran  f  e.  ran  f ) ) )
 
Theoremfin33i 7949* Inference from isfin3-3 7948. (This is actually a bit stronger than isfin3-3 7948 because it does not assume  F is a set and does not use the Axiom of Infinity either.) (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( A  e. FinIII  /\  F : om --> ~P A  /\  A. x  e.  om  ( F `
  suc  x )  C_  ( F `  x ) )  ->  |^| ran  F  e.  ran  F )
 
Theoremcompsscnvlem 7950* Lemma for compsscnv 7951. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( x  e. 
 ~P A  /\  y  =  ( A  \  x ) )  ->  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) )
 
Theoremcompsscnv 7951* Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  `' F  =  F
 
Theoremisf34lem1 7952* Lemma for isfin3-4 7962. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e.  V  /\  X  C_  A )  ->  ( F `  X )  =  ( A  \  X ) )
 
Theoremisf34lem2 7953* Lemma for isfin3-4 7962. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( A  e.  V  ->  F : ~P A --> ~P A )
 
Theoremcompssiso 7954* Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
 
Theoremisf34lem3 7955* Lemma for isfin3-4 7962. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( F "
 ( F " X ) )  =  X )
 
Theoremcompss 7956* Express image under of the complementation isomorphism. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( F " G )  =  { y  e.  ~P A  |  ( A  \  y )  e.  G }
 
Theoremisf34lem4 7957* Lemma for isfin3-4 7962. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. X )  =  |^| ( F " X ) )
 
Theoremisf34lem5 7958* Lemma for isfin3-4 7962. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  |^| X )  = 
 U. ( F " X ) )
 
Theoremisf34lem7 7959* Lemma for isfin3-4 7962. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e. FinIII  /\  G : om --> ~P A  /\  A. y  e.  om  ( G `
  y )  C_  ( G `  suc  y
 ) )  ->  U. ran  G  e.  ran  G )
 
Theoremisf34lem6 7960* Lemma for isfin3-4 7962. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( A  e.  V  ->  ( A  e. FinIII  <->  A. f  e.  ( ~P A  ^m  om )
 ( A. y  e.  om  ( f `  y
 )  C_  ( f `  suc  y )  ->  U. ran  f  e.  ran  f ) ) )
 
Theoremfin34i 7961* Inference from isfin3-4 7962. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( A  e. FinIII  /\  G : om --> ~P A  /\  A. x  e.  om  ( G `
  x )  C_  ( G `  suc  x ) )  ->  U. ran  G  e.  ran  G )
 
Theoremisfin3-4 7962* Weakly Dedekind-infinite sets are exactly those with an  om-indexed ascending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinIII  <->  A. f  e.  ( ~P A  ^m  om )
 ( A. x  e.  om  ( f `  x )  C_  ( f `  suc  x )  ->  U. ran  f  e.  ran  f ) ) )
 
Theoremfin11a 7963 Every I-finite set is Ia-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  Fin  ->  A  e. FinIa )
 
Theoremenfin1ai 7964 Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( A  e. FinIa  ->  B  e. FinIa ) )
 
Theoremisfin1-2 7965 A set is finite in the usual sense iff the power set of its power set is Dedekind finite. (Contributed by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  Fin  <->  ~P ~P A  e. FinIV )
 
Theoremisfin1-3 7966 A set is I-finite iff every system of subsets contains a maximal subset. Definition I of [Levy58] p. 2. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e.  Fin  <->  `' [ C.] 
 Fr  ~P A ) )
 
Theoremisfin1-4 7967 A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e.  Fin  <-> [ C.]  Fr 
 ~P A ) )
 
Theoremdffin1-5 7968 Compact quantifier-free version of the standard definition df-fin 6821. (Contributed by Stefan O'Rear, 6-Jan-2015.)
 |- 
 Fin  =  (  ~~  " om )
 
Theoremfin23 7969 Every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets). The proof here is the only one I could find, from http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm619.pdf p.94 (writeup by Tarski, credited to Kuratowski). Translated into English and modern notation, the proof proceeds as follows (variables renamed for uniqueness):

Suppose for a contradiction that  A is a set which is II-finite but not III-finite.

For any countable sequence of distinct subsets  T of  A, we can form a decreasing sequence of non-empty subsets  ( U `  T ) by taking finite intersections of initial segments of  T while skipping over any element of  T which would cause the intersection to be empty.

By II-finiteness (as fin2i2 7898) this sequence contains its intersection, call it  Y; since by induction every subset in the sequence  U is non-empty, the intersection must be non-empty.

Suppose that an element  X of  T has non-empty intersection with  Y. Thus said element has a non-empty intersection with the corresponding element of  U, therefore it was used in the construction of  U and all further elements of  U are subsets of  X, thus  X contains the  Y. That is, all elements of  X either contain  Y or are disjoint from it.

Since there are only two cases, there must exist an infinite subset of  T which uniformly either contain  Y or are disjoint from it. In the former case we can create an infinite set by subtracting  Y from each element. In either case, call the result  Z; this is an infinite set of subsets of 
A, each of which is disjoint from  Y and contained in the union of  T; the union of 
Z is strictly contained in the union of  T, because only the latter is a superset of the non-empty set  Y.

The preceeding four steps may be iterated a countable number of times starting from the assumed denumerable set of subsets to produce a denumerable sequence  B of the  T sets from each stage. Great caution is required to avoid ax-dc 8026 here; in particular an effective version of the pigeonhole principle (for aleph-null pigeons and 2 holes) is required. Since a denumerable set of subsets is assumed to exist, we can conclude  om  e.  _V without the axiom.

This  B sequence is strictly decreasing, thus it has no minimum, contradicting the first assumption. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

 |-  ( A  e. FinII  ->  A  e. FinIII )
 
Theoremfin34 7970 Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( A  e. FinIII  ->  A  e. FinIV )
 
Theoremisfin5-2 7971 Alternate definition of V-finite which emphasizes the idempotent behavior of V-infinite sets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinV  <->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A ) ) ) )
 
Theoremfin45 7972 Every IV-finite set is V-finite: if we can pack two copies of the set into itself, we can certainly leave space. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 18-May-2015.)
 |-  ( A  e. FinIV  ->  A  e. FinV
 )
 
Theoremfin56 7973 Every V-finite set is VI-finite because multiplication dominates addition for cardinals. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e. FinV  ->  A  e. FinVI )
 
Theoremfin17 7974 Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  Fin  ->  A  e. FinVII )
 
Theoremfin67 7975 Every VI-finite set is VII-finite. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e. FinVI  ->  A  e. FinVII )
 
Theoremisfin7-2 7976 A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinVII  <->  ( A  e.  dom  card  ->  A  e.  Fin ) ) )
 
Theoremfin71num 7977 A well-orderable set is VII-finite iff it is I-finite. Thus even without choice, on the class of well-orderable sets all eight definitions of finite set coincide. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  e.  dom  card 
 ->  ( A  e. FinVII  <->  A  e.  Fin ) )
 
Theoremdffin7-2 7978 Class form of isfin7-2 7976. (Contributed by Mario Carneiro, 17-May-2015.)
 |- FinVII  =  ( Fin  u.  ( _V  \  dom  card )
 )
 
Theoremdfacfin7 7979 Axiom of Choice equivalent: the VII-finite sets are the same as I-finite sets. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  (CHOICE  <-> FinVII  =  Fin )
 
Theoremfin1a2lem1 7980 Lemma for fin1a2 7995. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  S  =  ( x  e.  On  |->  suc  x )   =>    |-  ( A  e.  On  ->  ( S `  A )  =  suc  A )
 
Theoremfin1a2lem2 7981 Lemma for fin1a2 7995. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  S  =  ( x  e.  On  |->  suc  x )   =>    |-  S : On -1-1-> On
 
Theoremfin1a2lem3 7982 Lemma for fin1a2 7995. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   =>    |-  ( A  e.  om  ->  ( E `  A )  =  ( 2o  .o  A ) )
 
Theoremfin1a2lem4 7983 Lemma for fin1a2 7995. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   =>    |-  E : om -1-1-> om
 
Theoremfin1a2lem5 7984 Lemma for fin1a2 7995. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   =>    |-  ( A  e.  om  ->  ( A  e.  ran  E  <->  -. 
 suc  A  e.  ran  E ) )
 
Theoremfin1a2lem6 7985 Lemma for fin1a2 7995. Establish that  om can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   &    |-  S  =  ( x  e.  On  |->  suc  x )   =>    |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
 
Theoremfin1a2lem7 7986* Lemma for fin1a2 7995. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   &    |-  S  =  ( x  e.  On  |->  suc  x )   =>    |-  (
 ( A  e.  V  /\  A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \  y
 )  e. FinIII ) )  ->  A  e. FinIII )
 
Theoremfin1a2lem8 7987* Lemma for fin1a2 7995. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  ( ( A  e.  V  /\  A. x  e. 
 ~P  A ( x  e. FinIII  \/  ( A  \  x )  e. FinIII ) )  ->  A  e. FinIII )
 
Theoremfin1a2lem9 7988* Lemma for fin1a2 7995. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)
 |-  ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  { b  e.  X  |  b  ~<_  A }  e.  Fin )
 
Theoremfin1a2lem10 7989 Lemma for fin1a2 7995. A nonempty finite union of members of a chain is a member of the chain. (Contributed by Stefan O'Rear, 8-Nov-2014.)
 |-  ( ( A  =/=  (/)  /\  A  e.  Fin  /\ [ C.] 
 Or  A )  ->  U. A  e.  A )
 
Theoremfin1a2lem11 7990* Lemma for fin1a2 7995. (Contributed by Stefan O'Rear, 8-Nov-2014.)
 |-  ( ( [ C.]  Or  A  /\  A  C_  Fin )  ->  ran  (  b  e. 
 om  |->  U. { c  e.  A  |  c  ~<_  b } )  =  ( A  u.  { (/) } )
 )
 
Theoremfin1a2lem12 7991 Lemma for fin1a2 7995. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( ( ( A 
 C_  ~P B  /\ [ C.]  Or  A  /\  -.  U. A  e.  A )  /\  ( A  C_  Fin  /\  A  =/=  (/) ) )  ->  -.  B  e. FinIII )
 
Theoremfin1a2lem13 7992 Lemma for fin1a2 7995. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( ( ( A 
 C_  ~P B  /\ [ C.]  Or  A  /\  -.  U. A  e.  A )  /\  ( -.  C  e.  Fin  /\  C  e.  A )
 )  ->  -.  ( B  \  C )  e. FinII )
 
Theoremfin12 7993 Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 7995. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  Fin  ->  A  e. FinII )
 
Theoremfin1a2s 7994* An II-infinite set can have an I-infinite part broken off and remain II-infinite. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( ( A  e.  V  /\  A. x  e. 
 ~P  A ( x  e.  Fin  \/  ( A  \  x )  e. FinII ) )  ->  A  e. FinII )
 
Theoremfin1a2 7995 Every Ia-finite set is II-finite. Theorem 1 of [Levy58], p. 3. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e. FinIa  ->  A  e. FinII )
 
2.6.13  Hereditarily size-limited sets without Choice
 
Theoremitunifval 7996* Function value of iterated unions. EDITORIAL: The iterated unions and order types of ordered sets are split out here because they could concievably be independently useful. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  x )  |`  om )
 )   =>    |-  ( A  e.  V  ->  ( U `  A )  =  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  A )  |`  om )
 )
 
Theoremitunifn 7997* Functionality of the iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  x )  |`  om )
 )   =>    |-  ( A  e.  V  ->  ( U `  A )  Fn  om )
 
Theoremituni0 7998* A zero-fold iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  x )  |`  om )
 )   =>    |-  ( A  e.  V  ->  ( ( U `  A ) `  (/) )  =  A )
 
Theoremitunisuc 7999* Successor iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  x )  |`  om )
 )   =>    |-  ( ( U `  A ) `  suc  B )  =  U. (
 ( U `  A ) `  B )
 
Theoremitunitc1 8000* Each union iterate is a member of the transitive closure. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  x )  |`  om )
 )   =>    |-  ( ( U `  A ) `  B )  C_  ( TC `  A )
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