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Theorem List for Metamath Proof Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcfflb 7901* If there is a cofinal map from  B to  A, then  B is at least  ( cf `  A
). This theorem and cff1 7900 motivate the picture of  ( cf `  A
) as the greatest lower bound of the domain of cofinal maps into  A. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f
 ( f : B --> A  /\  A. z  e.  A  E. w  e.  B  z  C_  (
 f `  w )
 )  ->  ( cf `  A )  C_  B ) )
 
Theoremcfval2 7902* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( A  e.  On  ->  ( cf `  A )  =  |^|_ x  e. 
 { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  ( card `  x )
 )
 
Theoremcoflim 7903* A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( Lim  A  /\  B  C_  A )  ->  ( U. B  =  A 
 <-> 
 A. x  e.  A  E. y  e.  B  x  C_  y ) )
 
Theoremcflim3 7904* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  A  e.  _V   =>    |-  ( Lim  A  ->  ( cf `  A )  =  |^|_ x  e. 
 { x  e.  ~P A  |  U. x  =  A }  ( card `  x ) )
 
Theoremcflim2 7905 The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  A  e.  _V   =>    |-  ( Lim  A  <->  Lim  ( cf `  A ) )
 
Theoremcfom 7906 Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Proof shortened by Mario Carneiro, 11-Jun-2015.)
 |-  ( cf `  om )  =  om
 
Theoremcfss 7907* There is a cofinal subset of  A of cardinality  ( cf `  A
). (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( Lim  A  ->  E. x ( x 
 C_  A  /\  x  ~~  ( cf `  A )  /\  U. x  =  A ) )
 
Theoremcfslb 7908 Any cofinal subset of  A is at least as large as  ( cf `  A
). (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( Lim 
 A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A ) 
 ~<_  B )
 
Theoremcfslbn 7909 Any subset of  A smaller than its cofinality has union less than  A. (This is the contrapositive to cfslb 7908.) (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( Lim 
 A  /\  B  C_  A  /\  B  ~<  ( cf `  A ) )  ->  U. B  e.  A )
 
Theoremcfslb2n 7910* Any small collection of small subsets of  A cannot have union  A, where "small" means smaller than the cofinality. This is a stronger version of cfslb 7908. This is a common application of cofinality: under AC,  ( aleph `  1
) is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( Lim 
 A  /\  A. x  e.  B  ( x  C_  A  /\  x  ~<  ( cf `  A ) ) ) 
 ->  ( B  ~<  ( cf `  A )  ->  U. B  =/=  A ) )
 
Theoremcofsmo 7911* Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  C  =  { y  e.  B  |  A. w  e.  y  ( f `  w )  e.  (
 f `  y ) }   &    |-  K  =  |^| { x  e.  B  |  z  C_  ( f `  x ) }   &    |-  O  = OrdIso (  _E  ,  C )   =>    |-  ( ( Ord 
 A  /\  B  e.  On )  ->  ( E. f ( f : B --> A  /\  A. z  e.  A  E. w  e.  B  z  C_  (
 f `  w )
 )  ->  E. x  e.  suc  B E. g
 ( g : x --> A  /\  Smo  g  /\  A. z  e.  A  E. v  e.  x  z  C_  ( g `  v
 ) ) ) )
 
Theoremcfsmolem 7912* Lemma for cfsmo 7913. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  F  =  ( z  e.  _V  |->  ( ( g `  dom  z
 )  u.  U_ t  e.  dom  z  suc  (
 z `  t )
 ) )   &    |-  G  =  (recs ( F )  |`  ( cf `  A ) )   =>    |-  ( A  e.  On  ->  E. f ( f : ( cf `  A )
 --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z 
 C_  ( f `  w ) ) )
 
Theoremcfsmo 7913* The map in cff1 7900 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( A  e.  On  ->  E. f ( f : ( cf `  A )
 --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  ( cf `  A ) z 
 C_  ( f `  w ) ) )
 
Theoremcfcoflem 7914* Lemma for cfcof 7916, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f
 ( f : B --> A  /\  Smo  f  /\  A. x  e.  A  E. y  e.  B  x  C_  ( f `  y
 ) )  ->  ( cf `  A )  C_  ( cf `  B ) ) )
 
Theoremcoftr 7915* If there is a cofinal map from  B to  A and another from  C to  A, then there is also a cofinal map from  C to  B. Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 7916. (Contributed by Mario Carneiro, 16-Mar-2013.)
 |-  H  =  ( t  e.  C  |->  |^| { n  e.  B  |  ( g `
  t )  C_  ( f `  n ) } )   =>    |-  ( E. f ( f : B --> A  /\  Smo  f  /\  A. x  e.  A  E. y  e.  B  x  C_  (
 f `  y )
 )  ->  ( E. g ( g : C --> A  /\  A. z  e.  A  E. w  e.  C  z  C_  (
 g `  w )
 )  ->  E. h ( h : C --> B  /\  A. s  e.  B  E. w  e.  C  s  C_  ( h `  w ) ) ) )
 
Theoremcfcof 7916* If there is a cofinal map from  A to  B, then they have the same cofinality. This was used as Definition 11.1 of [TakeutiZaring] p. 100, who defines an equivalence relation cof  ( A ,  B ) and defines our  cf ( B ) as the minimum  B such that cof  ( A ,  B
). (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. f
 ( f : B --> A  /\  Smo  f  /\  A. z  e.  A  E. w  e.  B  z  C_  ( f `  w ) )  ->  ( cf `  A )  =  (
 cf `  B )
 ) )
 
Theoremcfidm 7917 The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( cf `  ( cf `  A ) )  =  ( cf `  A )
 
Theoremalephsing 7918 The cofinality of a limit aleph is the same as the cofinality of its argument, so if  ( aleph `  A )  <  A, then  ( aleph `  A
) is singular. Conversely, if  ( aleph `  A ) is regular (i.e. weakly inaccessible), then  ( aleph `  A )  =  A, so  A has to be rather large (see alephfp 7751). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  ( Lim  A  ->  (
 cf `  ( aleph `  A ) )  =  ( cf `  A ) )
 
2.6.12  Eight inequivalent definitions of finite set
 
Theoremsornom 7919* The range of a single-step monotone function from  om into a partially ordered set is a chain. (Contributed by Stefan O'Rear, 3-Nov-2014.)
 |-  ( ( F  Fn  om 
 /\  A. a  e.  om  ( ( F `  a ) R ( F `  suc  a
 )  \/  ( F `
  a )  =  ( F `  suc  a ) )  /\  R  Po  ran  F )  ->  R  Or  ran  F )
 
Syntaxcfin1a 7920 Extend class notation to include the class of Ia-finite sets.
 class FinIa
 
Syntaxcfin2 7921 Extend class notation to include the class of II-finite sets.
 class FinII
 
Syntaxcfin4 7922 Extend class notation to include the class of IV-finite sets.
 class FinIV
 
Syntaxcfin3 7923 Extend class notation to include the class of III-finite sets.
 class FinIII
 
Syntaxcfin5 7924 Extend class notation to include the class of V-finite sets.
 class FinV
 
Syntaxcfin6 7925 Extend class notation to include the class of VI-finite sets.
 class FinVI
 
Syntaxcfin7 7926 Extend class notation to include the class of VII-finite sets.
 class FinVII
 
Definitiondf-fin1a 7927* A set is Ia-finite iff it is not the union of two I-infinite sets. Equivalent to definition Ia of [Levy58] p. 2. A I-infinite Ia-finite set is also known as an amorphous set. This is the second of Levy's eight definitions of finite set. Levy's I-finite is equivalent to our df-fin 6883 and not repeated here. These eight definitions are equivalent with Choice but strictly decreasing in strength in models where Choice fails; conversely, they provide a series of increasingly stronger notions of infiniteness. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinIa  =  { x  |  A. y  e.  ~P  x ( y  e.  Fin  \/  ( x  \  y
 )  e.  Fin ) }
 
Definitiondf-fin2 7928* A set is II-finite (Tarski finite) iff every nonempty chain of subsets contains a maximum element. Definition II of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinII  =  { x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
 )  ->  U. y  e.  y ) }
 
Definitiondf-fin4 7929* A set is IV-finite (Dedekind finite) iff it has no equinumerous proper subset. Definition IV of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinIV  =  { x  |  -.  E. y ( y  C.  x  /\  y  ~~  x ) }
 
Definitiondf-fin3 7930 A set is III-finite (weakly Dedekind finite) iff its power set is Dedekind finite. Definition III of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinIII  =  { x  |  ~P x  e. FinIV }
 
Definitiondf-fin5 7931 A set is V-finite iff it behaves finitely under  +c. Definition V of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinV  =  { x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x ) ) }
 
Definitiondf-fin6 7932 A set is VI-finite iff it behaves finitely under  X.. Definition VI of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinVI  =  { x  |  ( x  ~<  2o  \/  x  ~<  ( x  X.  x ) ) }
 
Definitiondf-fin7 7933* A set is VII-finite iff it cannot be infinitely well-ordered. Equivalent to definition VII of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |- FinVII  =  { x  |  -.  E. y  e.  ( On  \  om ) x  ~~  y }
 
Theoremisfin1a 7934* Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinIa  <->  A. y  e.  ~P  A ( y  e. 
 Fin  \/  ( A  \  y )  e.  Fin ) ) )
 
Theoremfin1ai 7935 Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( ( A  e. FinIa  /\  X  C_  A )  ->  ( X  e.  Fin  \/  ( A  \  X )  e.  Fin ) )
 
Theoremisfin2 7936* Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinII  <->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
 )  ->  U. y  e.  y ) ) )
 
Theoremfin2i 7937 Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( ( ( A  e. FinII  /\  B  C_  ~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  U. B  e.  B )
 
Theoremisfin3 7938 Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e. FinIII  <->  ~P A  e. FinIV )
 
Theoremisfin4 7939* Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinIV  <->  -.  E. y ( y  C.  A  /\  y  ~~  A ) ) )
 
Theoremfin4i 7940 Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( ( X  C.  A  /\  X  ~~  A )  ->  -.  A  e. FinIV )
 
Theoremisfin5 7941 Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e. FinV  <->  ( A  =  (/) 
 \/  A  ~<  ( A  +c  A ) ) )
 
Theoremisfin6 7942 Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e. FinVI  <->  ( A  ~<  2o 
 \/  A  ~<  ( A  X.  A ) ) )
 
Theoremisfin7 7943* Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinVII  <->  -.  E. y  e.  ( On  \  om ) A  ~~  y ) )
 
Theoremsdom2en01 7944 A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( A  ~<  2o  <->  ( A  =  (/) 
 \/  A  ~~  1o ) )
 
Theoreminfpssrlem1 7945 Lemma for infpssr 7950. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  F : B -1-1-onto-> A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   &    |-  G  =  ( rec ( `' F ,  C )  |`  om )   =>    |-  ( ph  ->  ( G `  (/) )  =  C )
 
Theoreminfpssrlem2 7946 Lemma for infpssr 7950. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  F : B -1-1-onto-> A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   &    |-  G  =  ( rec ( `' F ,  C )  |`  om )   =>    |-  ( M  e.  om  ->  ( G `  suc  M )  =  ( `' F `  ( G `
  M ) ) )
 
Theoreminfpssrlem3 7947 Lemma for infpssr 7950. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  F : B -1-1-onto-> A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   &    |-  G  =  ( rec ( `' F ,  C )  |`  om )   =>    |-  ( ph  ->  G : om --> A )
 
Theoreminfpssrlem4 7948 Lemma for infpssr 7950. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  F : B -1-1-onto-> A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   &    |-  G  =  ( rec ( `' F ,  C )  |`  om )   =>    |-  ( ( ph  /\  M  e.  om  /\  N  e.  M )  ->  ( G `
  M )  =/=  ( G `  N ) )
 
Theoreminfpssrlem5 7949 Lemma for infpssr 7950. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ph  ->  F : B -1-1-onto-> A )   &    |-  ( ph  ->  C  e.  ( A  \  B ) )   &    |-  G  =  ( rec ( `' F ,  C )  |`  om )   =>    |-  ( ph  ->  ( A  e.  V  ->  om  ~<_  A ) )
 
Theoreminfpssr 7950 Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
 |-  ( ( X  C.  A  /\  X  ~~  A )  ->  om  ~<_  A )
 
Theoremfin4en1 7951 Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
 |-  ( A  ~~  B  ->  ( A  e. FinIV  ->  B  e. FinIV ) )
 
Theoremssfin4 7952 Dedekind finite sets have Dedekind finite subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( ( A  e. FinIV  /\  B  C_  A )  ->  B  e. FinIV )
 
Theoremdomfin4 7953 A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.)
 |-  ( ( A  e. FinIV  /\  B 
 ~<_  A )  ->  B  e. FinIV )
 
Theoremominf4 7954  om is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
 |- 
 -.  om  e. FinIV
 
TheoreminfpssALT 7955* A set with a denumerable subset has a proper subset equinumerous to it, proved without AC or Infinity. Unlike infpss 7859, it uses Replacement. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( om  ~<_  A  ->  E. x ( x  C.  A  /\  x  ~~  A ) )
 
Theoremisfin4-2 7956 Alternate definition of IV-finite sets: they lack a denumerable subset. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinIV  <->  -.  om  ~<_  A ) )
 
Theoremisfin4-3 7957 Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 7939 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  e. FinIV  <->  A  ~<  ( A  +c  1o ) )
 
Theoremfin23lem7 7958* Lemma for isfin2-2 7961. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
 |-  ( ( A  e.  V  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  ( A  \  x )  e.  B }  =/=  (/) )
 
Theoremfin23lem11 7959* Lemma for isfin2-2 7961. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
 |-  ( z  =  ( A  \  x ) 
 ->  ( ps  <->  ch ) )   &    |-  ( w  =  ( A  \  v )  ->  ( ph 
 <-> 
 th ) )   &    |-  (
 ( x  C_  A  /\  v  C_  A ) 
 ->  ( ch  <->  th ) )   =>    |-  ( B  C_  ~P A  ->  ( E. x  e.  { c  e.  ~P A  |  ( A  \  c )  e.  B } A. w  e.  { c  e.  ~P A  |  ( A  \  c )  e.  B }  -.  ph 
 ->  E. z  e.  B  A. v  e.  B  -.  ps ) )
 
Theoremfin2i2 7960 A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)
 |-  ( ( ( A  e. FinII  /\  B  C_  ~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  |^| B  e.  B )
 
Theoremisfin2-2 7961* FinII expressed in terms of minimal elements. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinII  <->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
 )  ->  |^| y  e.  y ) ) )
 
Theoremssfin2 7962 A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.)
 |-  ( ( A  e. FinII  /\  B  C_  A )  ->  B  e. FinII )
 
Theoremenfin2i 7963 II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( A  e. FinII  ->  B  e. FinII ) )
 
Theoremfin23lem24 7964 Lemma for fin23 8031. 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 7965 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 7966 Lemma for fin23 8031. 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 7967* Lemma for fin23lem22 7969. (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 7968* Lemma for fin23lem22 7969. (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 7969* Lemma for fin23 8031 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 7970) 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 7970* The mapping constructed in fin23lem22 7969 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 7971* 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 7972* 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 7973* 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 7974* Lemma for fin23 8031. 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 7975* Lemma for fin23 8031. 
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 7976* Lemma for fin23 8031. 
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 7977* Lemma for fin23 8031. 
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 7978* Lemma for fin23 8031. 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 7979* Lemma for fin23 8031. 
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 7980* Lemma for fin23 8031. 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 7981* Lemma for fin23 8031. 
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 7982* Lemma for fin23 8031. 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 7983* Lemma for fin23 8031. 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 7984* Lemma for fin23 8031. 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 7985* Lemma for fin23 8031. 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 7986* Lemma for fin23 8031. 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 7987* Lemma for fin23 8031. 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 7988* Lemma for fin23 8031. 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 7989* Lemma for fin23 8031. 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 7990* Lemma for fin23 8031. 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 7991* Lemma for fin23 8031. 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 7992* Lemma for fin23 8031. 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 7993* Lemma for fin23 8031. 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 7994* Lemma for fin23 8031. 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 7995* Lemma for isfin3-2 8009. 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 7996* Lemma for isfin3-2 8009. 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 7997* Lemma for isfin3-2 8009. 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 7998* Lemma for isfin3-2 8009. 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 7999* Lemma for isfin3-2 8009. 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 8000* Lemma for isfin3-2 8009. 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 )  =/=  (/) )
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