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Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopth1 4001 Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  ->  A  =  C )
 
Theoremopth 4002 The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that  C and  D are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
 ( A  =  C  /\  B  =  D ) )
 
Theoremopthg 4003 Ordered pair theorem.  C and  D are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >. 
 <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremopthg2 4004 Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >. 
 <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremopth2 4005 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
 |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
 ( A  =  C  /\  B  =  D ) )
 
Theoremotth2 4006 Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  e.  _V   =>    |-  ( <.
 <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
 
Theoremotth 4007 Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  e.  _V   =>    |-  ( <. A ,  B ,  R >.  =  <. C ,  D ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S )
 )
 
Theoremeqvinop 4008* A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  =  <. B ,  C >.  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. ) )
 
Theoremcopsexg 4009* Substitution of class  A for ordered pair  <. x ,  y
>.. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  ( A  =  <. x ,  y >.  ->  ( ph 
 <-> 
 E. x E. y
 ( A  =  <. x ,  y >.  /\  ph )
 ) )
 
Theoremcopsex2t 4010* Closed theorem form of copsex2g 4011. (Contributed by NM, 17-Feb-2013.)
 |-  ( ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )  /\  ( A  e.  V  /\  B  e.  W ) )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph )  <->  ps ) )
 
Theoremcopsex2g 4011* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y (
 <. A ,  B >.  = 
 <. x ,  y >.  /\  ph )  <->  ps ) )
 
Theoremcopsex4g 4012* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
 |-  ( ( ( x  =  A  /\  y  =  B )  /\  (
 z  =  C  /\  w  =  D )
 )  ->  ( ph  <->  ps ) )   =>    |-  ( ( ( A  e.  R  /\  B  e.  S )  /\  ( C  e.  R  /\  D  e.  S )
 )  ->  ( E. x E. y E. z E. w ( ( <. A ,  B >.  =  <. x ,  y >.  /\  <. C ,  D >.  =  <. z ,  w >. )  /\  ph )  <->  ps ) )
 
Theorem0nelop 4013 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 -.  (/)  e.  <. A ,  B >.
 
Theoremopeqex 4014 Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
 |-  ( <. A ,  B >.  =  <. C ,  D >.  ->  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( C  e.  _V 
 /\  D  e.  _V ) ) )
 
Theoremopcom 4015 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 4016* "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 4017 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 4018 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 4019* 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 4020 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 4021 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 )
 
2.3.4  Ordered-pair class abstractions (cont.)
 
Theoremopabid 4022 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 4023* 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 4024* The law of concretion in terms of substitutions. Less general than opelopabsb 4025, 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 4025* 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 4026* 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 4027* Closed theorem form of opelopab 4036. (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 4028* 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 4029* 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 4030* 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 4031* 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 4032* 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 4033* 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 4034* 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 4035* 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 4036* 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 4037* 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 4038* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4036 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 4039* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4036 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 4040 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 4041 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 4042 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
 |-  ( ph  ->  ps )   =>    |-  { <. x ,  y >.  |  ph } 
 C_  { <. x ,  y >.  |  ps }
 
Theoremssopab2dv 4043* 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 4044 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 4045* 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 4046* 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 4047 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 4048 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 4049 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 4050 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 4051 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 4052 Extend class notation to include the epsilon relation.
 class  _E
 
Syntaxcid 4053 Extend the definition of a class to include identity relation.
 class  _I
 
Definitiondf-eprel 4054* 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 4055. Thus, 5  _E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)
 |- 
 _E  =  { <. x ,  y >.  |  x  e.  y }
 
Theoremepelg 4055 The epsilon relation and membership are the same. General version of epel 4057. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
 
Theoremepelc 4056 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 4057 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
 |-  ( x  _E  y  <->  x  e.  y )
 
Definitiondf-id 4058* 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 4059 Extend wff notation to include the strict partial ordering predicate. Read: '  R is a partial order on  A.'
 wff  R  Po  A
 
Syntaxwor 4060 Extend wff notation to include the strict linear ordering predicate. Read: '  R orders  A.'
 wff  R  Or  A
 
Definitiondf-po 4061* 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 4062* 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 4063 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 4064 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
 
Theorempoeq2 4065 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
 
Theoremnfpo 4066 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 4067 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 4068 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 4069* 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 4070* 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 4071* 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 4072 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
 
Theorempotr 4073 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 4074 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 4075 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 4076 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 4077* 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 4078 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
 |-  ( R  Or  A  ->  R  Po  A )
 
Theoremsoss 4079 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 4080 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )
 
Theoremsoeq2 4081 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Or  A  <->  R  Or  B ) )
 
Theoremsonr 4082 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
 |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
 
Theoremsotr 4083 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 4084* An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4062). (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 4085 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 4086 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 4087 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 4088 One direction of sotritric 4089 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 4089 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 4090 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 4091 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 4092 Extend wff notation to include the well-founded predicate.
 wff FrFor  R A S
 
Syntaxwfr 4093 Extend wff notation to include the well-founded predicate. Read: '  R is a well-founded relation on 
A.'
 wff  R  Fr  A
 
Syntaxwse 4094 Extend wff notation to include the set-like predicate. Read: '  R is set-like on  A.'
 wff  R Se  A
 
Syntaxwwe 4095 Extend wff notation to include the well-ordering predicate. Read: '  R well-orders  A.'
 wff  R  We  A
 
Definitiondf-frfor 4096* 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 4097* 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 4098* 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 4099* 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 4275). 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 4100* 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 )
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