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Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembrabga 4001* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
 
Theoremopelopab2a 4002* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) } 
 <->  ps ) )
 
Theoremopelopaba 4003* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ps )
 
Theorembraba 4004* The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( A R B 
 <->  ps )
 
Theoremopelopabg 4005* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ph }  <->  ch ) )
 
Theorembrabg 4006* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
 
Theoremopelopab2 4007* Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) } 
 <->  ch ) )
 
Theoremopelopab 4008* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
 
Theorembrab 4009* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   &    |-  R  =  { <. x ,  y >.  | 
 ph }   =>    |-  ( A R B  <->  ch )
 
Theoremopelopabaf 4010* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4008 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |- 
 F/ x ps   &    |-  F/ y ps   &    |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ps )
 
Theoremopelopabf 4011* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4008 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.)
 |- 
 F/ x ps   &    |-  F/ y ch   &    |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
 
Theoremssopab2 4012 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
 |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  {
 <. x ,  y >.  |  ps } )
 
Theoremssopab2b 4013 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  ( { <. x ,  y >.  |  ph }  C_  {
 <. x ,  y >.  |  ps }  <->  A. x A. y
 ( ph  ->  ps )
 )
 
Theoremssopab2i 4014 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
 |-  ( ph  ->  ps )   =>    |-  { <. x ,  y >.  |  ph } 
 C_  { <. x ,  y >.  |  ps }
 
Theoremssopab2dv 4015* Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ps }  C_  { <. x ,  y >.  |  ch } )
 
Theoremeqopab2b 4016 Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }  <->  A. x A. y
 ( ph  <->  ps ) )
 
Theoremopabm 4017* Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.)
 |-  ( E. z  z  e.  { <. x ,  y >.  |  ph }  <->  E. x E. y ph )
 
Theoremiunopab 4018* Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  U_ z  e.  A  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  E. z  e.  A  ph }
 
2.3.5  Power class of union and intersection
 
Theorempwin 4019 The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
 |- 
 ~P ( A  i^i  B )  =  ( ~P A  i^i  ~P B )
 
Theorempwunss 4020 The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
 |-  ( ~P A  u.  ~P B )  C_  ~P ( A  u.  B )
 
Theorempwssunim 4021 The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.)
 |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B ) )
 
Theorempwundifss 4022 Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.)
 |-  ( ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )  C_  ~P ( A  u.  B )
 
Theorempwunim 4023 The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by Jim Kingdon, 30-Sep-2018.)
 |-  ( ( A  C_  B  \/  B  C_  A )  ->  ~P ( A  u.  B )  =  ( ~P A  u.  ~P B ) )
 
2.3.6  Epsilon and identity relations
 
Syntaxcep 4024 Extend class notation to include the epsilon relation.
 class  _E
 
Syntaxcid 4025 Extend the definition of a class to include identity relation.
 class  _I
 
Definitiondf-eprel 4026* Define the epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is,  ( A  _E  B  <->  A  e.  B ) when  B is a set by epelg 4027. Thus, 5  _E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)
 |- 
 _E  =  { <. x ,  y >.  |  x  e.  y }
 
Theoremepelg 4027 The epsilon relation and membership are the same. General version of epel 4029. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
 
Theoremepelc 4028 The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  B  e.  _V   =>    |-  ( A  _E  B 
 <->  A  e.  B )
 
Theoremepel 4029 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
 |-  ( x  _E  y  <->  x  e.  y )
 
Definitiondf-id 4030* Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5  _I 5 and  -. 4  _I 5. (Contributed by NM, 13-Aug-1995.)
 |- 
 _I  =  { <. x ,  y >.  |  x  =  y }
 
2.3.7  Partial and complete ordering
 
Syntaxwpo 4031 Extend wff notation to include the strict partial ordering predicate. Read: '  R is a partial order on  A.'
 wff  R  Po  A
 
Syntaxwor 4032 Extend wff notation to include the strict linear ordering predicate. Read: '  R orders  A.'
 wff  R  Or  A
 
Definitiondf-po 4033* Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression  R  Po  A means  R is a partial order on  A. (Contributed by NM, 16-Mar-1997.)
 |-  ( R  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
 
Definitiondf-iso 4034* Define the strict linear order predicate. The expression  R  Or  A is true if relationship  R orders  A. The property  x R y  ->  ( x R z  \/  z R y ) is called weak linearity by Proposition 11.2.3 of [HoTT], p. (varies). If we assumed excluded middle, it would be equivalent to trichotomy, 
x R y  \/  x  =  y  \/  y R x. (Contributed by NM, 21-Jan-1996.) (Revised by Jim Kingdon, 4-Oct-2018.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x R y  ->  ( x R z  \/  z R y ) ) ) )
 
Theoremposs 4035 Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  ( A  C_  B  ->  ( R  Po  B  ->  R  Po  A ) )
 
Theorempoeq1 4036 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
 
Theorempoeq2 4037 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
 
Theoremnfpo 4038 Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Po  A
 
Theoremnfso 4039 Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Or  A
 
Theorempocl 4040 Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  Po  A  ->  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) 
 ->  ( -.  B R B  /\  ( ( B R C  /\  C R D )  ->  B R D ) ) ) )
 
Theoremispod 4041* Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  -.  x R x )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  ( ( x R y  /\  y R z )  ->  x R z ) )   =>    |-  ( ph  ->  R  Po  A )
 
Theoremswopolem 4042* Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )
 )  ->  ( x R y  ->  ( x R z  \/  z R y ) ) )   =>    |-  ( ( ph  /\  ( X  e.  A  /\  Y  e.  A  /\  Z  e.  A )
 )  ->  ( X R Y  ->  ( X R Z  \/  Z R Y ) ) )
 
Theoremswopo 4043* A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( ph  /\  (
 y  e.  A  /\  z  e.  A )
 )  ->  ( y R z  ->  -.  z R y ) )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )
 )  ->  ( x R y  ->  ( x R z  \/  z R y ) ) )   =>    |-  ( ph  ->  R  Po  A )
 
Theorempoirr 4044 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
 
Theorempotr 4045 A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  ( ( B R C  /\  C R D )  ->  B R D ) )
 
Theorempo2nr 4046 A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  -.  ( B R C  /\  C R B ) )
 
Theorempo3nr 4047 A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )
 
Theorempo0 4048 Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  R  Po  (/)
 
Theorempofun 4049* A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)
 |-  S  =  { <. x ,  y >.  |  X R Y }   &    |-  ( x  =  y  ->  X  =  Y )   =>    |-  ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  ->  S  Po  A )
 
Theoremsopo 4050 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
 |-  ( R  Or  A  ->  R  Po  A )
 
Theoremsoss 4051 Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
 
Theoremsoeq1 4052 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )
 
Theoremsoeq2 4053 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Or  A  <->  R  Or  B ) )
 
Theoremsonr 4054 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
 |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
 
Theoremsotr 4055 A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  ( ( B R C  /\  C R D )  ->  B R D ) )
 
Theoremissod 4056* An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4034). (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Po  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A ) )  ->  ( x R y  \/  x  =  y  \/  y R x ) )   =>    |-  ( ph  ->  R  Or  A )
 
Theoremsowlin 4057 A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  ( B R C  ->  ( B R D  \/  D R C ) ) )
 
Theoremso2nr 4058 A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  -.  ( B R C  /\  C R B ) )
 
Theoremso3nr 4059 A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )
 
Theoremsotricim 4060 One direction of sotritric 4061 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  ( B R C  ->  -.  ( B  =  C  \/  C R B ) ) )
 
Theoremsotritric 4061 A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  R  Or  A   &    |-  (
 ( B  e.  A  /\  C  e.  A ) 
 ->  ( B R C  \/  B  =  C  \/  C R B ) )   =>    |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B R C 
 <->  -.  ( B  =  C  \/  C R B ) ) )
 
Theoremsotritrieq 4062 A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |-  R  Or  A   &    |-  (
 ( B  e.  A  /\  C  e.  A ) 
 ->  ( B R C  \/  B  =  C  \/  C R B ) )   =>    |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B  =  C 
 <->  -.  ( B R C  \/  C R B ) ) )
 
Theoremso0 4063 Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  R  Or  (/)
 
2.3.8  Founded and set-like relations
 
Syntaxwfrfor 4064 Extend wff notation to include the well-founded predicate.
 wff FrFor  R A S
 
Syntaxwfr 4065 Extend wff notation to include the well-founded predicate. Read: '  R is a well-founded relation on 
A.'
 wff  R  Fr  A
 
Syntaxwse 4066 Extend wff notation to include the set-like predicate. Read: '  R is set-like on  A.'
 wff  R Se  A
 
Syntaxwwe 4067 Extend wff notation to include the well-ordering predicate. Read: '  R well-orders  A.'
 wff  R  We  A
 
Definitiondf-frfor 4068* Define the well-founded relation predicate where  A might be a proper class. By passing in  S we allow it potentially to be a proper class rather than a set. (Contributed by Jim Kingdon and Mario Carneiro, 22-Sep-2021.)
 |-  (FrFor  R A S  <->  (
 A. x  e.  A  ( A. y  e.  A  ( y R x 
 ->  y  e.  S )  ->  x  e.  S )  ->  A  C_  S ) )
 
Definitiondf-frind 4069* Define the well-founded relation predicate. In the presence of excluded middle, there are a variety of equivalent ways to define this. In our case, this definition, in terms of an inductive principle, works better than one along the lines of "there is an element which is minimal when A is ordered by R". Because  s is constrained to be a set (not a proper class) here, sometimes it may be necessary to use FrFor directly rather than via  Fr. (Contributed by Jim Kingdon and Mario Carneiro, 21-Sep-2021.)
 |-  ( R  Fr  A  <->  A. sFrFor  R A s )
 
Definitiondf-se 4070* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
 |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
 
Definitiondf-wetr 4071* Define the well-ordering predicate. It is unusual to define "well-ordering" in the absence of excluded middle, but we mean an ordering which is like the ordering which we have for ordinals (for example, it does not entail trichotomy because ordinals don't have that as seen at ordtriexmid 4247). Given excluded middle, well-ordering is usually defined to require trichotomy (and the defintion of  Fr is typically also different). (Contributed by Mario Carneiro and Jim Kingdon, 23-Sep-2021.)
 |-  ( R  We  A  <->  ( R  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z ) ) )
 
Theoremseex 4072* The  R-preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)
 |-  ( ( R Se  A  /\  B  e.  A ) 
 ->  { x  e.  A  |  x R B }  e.  _V )
 
Theoremexse 4073 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( A  e.  V  ->  R Se  A )
 
Theoremsess1 4074 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  C_  S  ->  ( S Se  A  ->  R Se 
 A ) )
 
Theoremsess2 4075 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  C_  B  ->  ( R Se  B  ->  R Se 
 A ) )
 
Theoremseeq1 4076 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
 )
 
Theoremseeq2 4077 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
 )
 
Theoremnfse 4078 Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R Se  A
 
Theoremepse 4079 The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.)
 |- 
 _E Se  A
 
Theoremfrforeq1 4080 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
 |-  ( R  =  S  ->  (FrFor  R A T  <-> FrFor  S A T ) )
 
Theoremfreq1 4081 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
 
Theoremfrforeq2 4082 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
 |-  ( A  =  B  ->  (FrFor  R A T  <-> FrFor  R B T ) )
 
Theoremfreq2 4083 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
 |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
 
Theoremfrforeq3 4084 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
 |-  ( S  =  T  ->  (FrFor  R A S  <-> FrFor  R A T ) )
 
Theoremnffrfor 4085 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   &    |-  F/_ x S   =>    |- 
 F/ xFrFor  R A S
 
Theoremnffr 4086 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Fr  A
 
Theoremfrirrg 4087 A well-founded relation is irreflexive. This is the case where  A exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
 |-  ( ( R  Fr  A  /\  A  e.  V  /\  B  e.  A ) 
 ->  -.  B R B )
 
Theoremfr0 4088 Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
 |-  R  Fr  (/)
 
Theoremfrind 4089* Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  (
 ( ch  /\  x  e.  A )  ->  ( A. y  e.  A  ( y R x 
 ->  ps )  ->  ph )
 )   &    |-  ( ch  ->  R  Fr  A )   &    |-  ( ch  ->  A  e.  V )   =>    |-  ( ( ch 
 /\  x  e.  A )  ->  ph )
 
Theoremefrirr 4090 Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
 |-  (  _E  Fr  A  ->  -.  A  e.  A )
 
Theoremtz7.2 4091 Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent  _E  Fr  A. (Contributed by NM, 4-May-1994.)
 |-  ( ( Tr  A  /\  _E  Fr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
 
Theoremnfwe 4092 Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  We  A
 
Theoremweeq1 4093 Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
 |-  ( R  =  S  ->  ( R  We  A  <->  S  We  A ) )
 
Theoremweeq2 4094 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
 |-  ( A  =  B  ->  ( R  We  A  <->  R  We  B ) )
 
Theoremwefr 4095 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
 |-  ( R  We  A  ->  R  Fr  A )
 
Theoremwepo 4096 A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
 |-  ( ( R  We  A  /\  A  e.  V )  ->  R  Po  A )
 
Theoremwetrep 4097* An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
 |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  ( ( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
 
Theoremwe0 4098 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
 |-  R  We  (/)
 
2.3.9  Ordinals
 
Syntaxword 4099 Extend the definition of a wff to include the ordinal predicate.
 wff  Ord  A
 
Syntaxcon0 4100 Extend the definition of a class to include the class of all ordinal numbers. (The 0 in the name prevents creating a file called con.html, which causes problems in Windows.)
 class  On
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