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Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopcom 4101 An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  =  <. B ,  A >.  <->  A  =  B )
 
Theoremmoop2 4102* "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  B  e.  _V   =>    |-  E* x  A  =  <. B ,  x >.
 
Theoremopeqsn 4103 Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )
 
Theoremopeqpr 4104 Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <. A ,  B >.  =  { C ,  D }  <->  ( ( C  =  { A }  /\  D  =  { A ,  B } )  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
 
Theoremeuotd 4105* Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  ( ps 
 <->  ( a  =  A  /\  b  =  B  /\  c  =  C ) ) )   =>    |-  ( ph  ->  E! x E. a E. b E. c ( x  =  <. a ,  b ,  c >.  /\  ps )
 )
 
Theoremuniop 4106 The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. <. A ,  B >.  =  { A ,  B }
 
Theoremuniopel 4107 Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  C  ->  U. <. A ,  B >.  e.  U. C )
 
2.3.5  Ordered-pair class abstractions (cont.)
 
Theoremopabid 4108 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( <. x ,  y >.  e.  { <. x ,  y >.  |  ph }  <->  ph )
 
Theoremelopab 4109* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
 |-  ( A  e.  { <. x ,  y >.  | 
 ph }  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  ph )
 )
 
TheoremopelopabsbALT 4110* The law of concretion in terms of substitutions. Less general than opelopabsb 4111, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( <. z ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ w  /  y ] [ z  /  x ] ph )
 
Theoremopelopabsb 4111* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
 |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ].
 [. B  /  y ]. ph )
 
Theorembrabsb 4112* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
 |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( A R B  <->  [. A  /  x ].
 [. B  /  y ]. ph )
 
Theoremopelopabt 4113* Closed theorem form of opelopab 4122. (Contributed by NM, 19-Feb-2013.)
 |-  ( ( A. x A. y ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x A. y ( y  =  B  ->  ( ps  <->  ch ) )  /\  ( A  e.  V  /\  B  e.  W ) )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch ) )
 
Theoremopelopabga 4114* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ph }  <->  ps ) )
 
Theorembrabga 4115* 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 4116* 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 4117* 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 4118* 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 4119* 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 4120* 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 4121* 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 4122* 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 4123* 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 4124* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4122 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 4125* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4122 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 4126 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 4127 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 4128 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
 |-  ( ph  ->  ps )   =>    |-  { <. x ,  y >.  |  ph } 
 C_  { <. x ,  y >.  |  ps }
 
Theoremssopab2dv 4129* 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 4130 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 4131* 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 4132* 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 }
 
2.3.6  Power class of union and intersection
 
Theorempwin 4133 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 4134 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 4135 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 4136 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 4137 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 ) )
 
2.3.7  Epsilon and identity relations
 
Syntaxcep 4138 Extend class notation to include the epsilon relation.
 class  _E
 
Syntaxcid 4139 Extend the definition of a class to include identity relation.
 class  _I
 
Definitiondf-eprel 4140* 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 4141. Thus, 5  _E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)
 |- 
 _E  =  { <. x ,  y >.  |  x  e.  y }
 
Theoremepelg 4141 The epsilon relation and membership are the same. General version of epel 4143. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
 
Theoremepelc 4142 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 4143 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
 |-  ( x  _E  y  <->  x  e.  y )
 
Definitiondf-id 4144* 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 }
 
2.3.8  Partial and complete ordering
 
Syntaxwpo 4145 Extend wff notation to include the strict partial ordering predicate. Read: '  R is a partial order on  A.'
 wff  R  Po  A
 
Syntaxwor 4146 Extend wff notation to include the strict linear ordering predicate. Read: '  R orders  A.'
 wff  R  Or  A
 
Definitiondf-po 4147* 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 4148* 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 4149 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 4150 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
 
Theorempoeq2 4151 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
 
Theoremnfpo 4152 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 4153 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 4154 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 4155* 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 4156* 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 4157* 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 4158 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
 
Theorempotr 4159 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 4160 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 4161 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 4162 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 4163* 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 4164 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
 |-  ( R  Or  A  ->  R  Po  A )
 
Theoremsoss 4165 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 4166 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )
 
Theoremsoeq2 4167 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Or  A  <->  R  Or  B ) )
 
Theoremsonr 4168 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
 |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
 
Theoremsotr 4169 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 4170* An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4148). (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 4171 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 4172 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 4173 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 4174 One direction of sotritric 4175 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 4175 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 4176 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 4177 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  (/)
 
2.3.9  Founded and set-like relations
 
Syntaxwfrfor 4178 Extend wff notation to include the well-founded predicate.
 wff FrFor  R A S
 
Syntaxwfr 4179 Extend wff notation to include the well-founded predicate. Read: '  R is a well-founded relation on 
A.'
 wff  R  Fr  A
 
Syntaxwse 4180 Extend wff notation to include the set-like predicate. Read: '  R is set-like on  A.'
 wff  R Se  A
 
Syntaxwwe 4181 Extend wff notation to include the well-ordering predicate. Read: '  R well-orders  A.'
 wff  R  We  A
 
Definitiondf-frfor 4182* 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 4183* 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 4184* 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 4185* 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 4366). 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 4186* 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 4187 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( A  e.  V  ->  R Se  A )
 
Theoremsess1 4188 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  C_  S  ->  ( S Se  A  ->  R Se 
 A ) )
 
Theoremsess2 4189 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  C_  B  ->  ( R Se  B  ->  R Se 
 A ) )
 
Theoremseeq1 4190 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( R  =  S  ->  ( R Se  A  <->  S Se  A )
 )
 
Theoremseeq2 4191 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  =  B  ->  ( R Se  A  <->  R Se  B )
 )
 
Theoremnfse 4192 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 4193 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 4194 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 4195 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
 
Theoremfrforeq2 4196 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 4197 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
 |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
 
Theoremfrforeq3 4198 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 4199 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 4200 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
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