P. 41 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	copsex2g 4001*	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 ) )
Theorem	copsex4g 4002*	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 ) )
Theorem	0nelop 4003	A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- -. (/) e. <. A , B >.
Theorem	opeqex 4004	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 ) ) )
Theorem	opcom 4005	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 )
Theorem	moop2 4006*	"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 >.
Theorem	opeqsn 4007	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 } ) )
Theorem	opeqpr 4008	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 } ) ) )
Theorem	euotd 4009*	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 ) )
Theorem	uniop 4010	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 }
Theorem	uniopel 4011	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.)
Theorem	opabid 4012	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 )
Theorem	elopab 4013*	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 ) )
Theorem	opelopabsbALT 4014*	The law of concretion in terms of substitutions. Less general than opelopabsb 4015, 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 )
Theorem	opelopabsb 4015*	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 )
Theorem	brabsb 4016*	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 )
Theorem	opelopabt 4017*	Closed theorem form of opelopab 4026. (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 ) )
Theorem	opelopabga 4018*	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 ) )
Theorem	brabga 4019*	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 ) )
Theorem	opelopab2a 4020*	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 ) )
Theorem	opelopaba 4021*	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 )
Theorem	braba 4022*	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 )
Theorem	opelopabg 4023*	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 ) )
Theorem	brabg 4024*	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 ) )
Theorem	opelopab2 4025*	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 ) )
Theorem	opelopab 4026*	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 )
Theorem	brab 4027*	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 )
Theorem	opelopabaf 4028*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4026 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 )
Theorem	opelopabf 4029*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4026 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 )
Theorem	ssopab2 4030	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 } )
Theorem	ssopab2b 4031	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 ) )
Theorem	ssopab2i 4032	Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
|- ( ph -> ps )   =>    |- { <. x , y >. | ph } C_ { <. x , y >. | ps }
Theorem	ssopab2dv 4033*	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 } )
Theorem	eqopab2b 4034	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 ) )
Theorem	opabm 4035*	Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.)
|- ( E. z z e. { <. x , y >. | ph } <-> E. x E. y ph )
Theorem	iunopab 4036*	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
Theorem	pwin 4037	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 )
Theorem	pwunss 4038	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 )
Theorem	pwssunim 4039	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 ) )
Theorem	pwundifss 4040	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 )
Theorem	pwunim 4041	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
Syntax	cep 4042	Extend class notation to include the epsilon relation.
class _E
Syntax	cid 4043	Extend the definition of a class to include identity relation.
class _I
Definition	df-eprel 4044*	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 4045. Thus, 5 _E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)
|- _E = { <. x , y >. | x e. y }
Theorem	epelg 4045	The epsilon relation and membership are the same. General version of epel 4047. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( B e. V -> ( A _E B <-> A e. B ) )
Theorem	epelc 4046	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 )
Theorem	epel 4047	The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
|- ( x _E y <-> x e. y )
Definition	df-id 4048*	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
Syntax	wpo 4049	Extend wff notation to include the strict partial ordering predicate. Read: ' R is a partial order on A.'
wff R Po A
Syntax	wor 4050	Extend wff notation to include the strict linear ordering predicate. Read: ' R orders A.'
wff R Or A
Definition	df-po 4051*	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 ) ) )
Definition	df-iso 4052*	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 ) ) ) )
Theorem	poss 4053	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 ) )
Theorem	poeq1 4054	Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
|- ( R = S -> ( R Po A <-> S Po A ) )
Theorem	poeq2 4055	Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
|- ( A = B -> ( R Po A <-> R Po B ) )
Theorem	nfpo 4056	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
Theorem	nfso 4057	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
Theorem	pocl 4058	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 ) ) ) )
Theorem	ispod 4059*	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 )
Theorem	swopolem 4060*	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 ) ) )
Theorem	swopo 4061*	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 )
Theorem	poirr 4062	A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
|- ( ( R Po A /\ B e. A ) -> -. B R B )
Theorem	potr 4063	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 ) )
Theorem	po2nr 4064	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 ) )
Theorem	po3nr 4065	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 ) )
Theorem	po0 4066	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 (/)
Theorem	pofun 4067*	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 )
Theorem	sopo 4068	A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
|- ( R Or A -> R Po A )
Theorem	soss 4069	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 ) )
Theorem	soeq1 4070	Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
|- ( R = S -> ( R Or A <-> S Or A ) )
Theorem	soeq2 4071	Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
|- ( A = B -> ( R Or A <-> R Or B ) )
Theorem	sonr 4072	A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
|- ( ( R Or A /\ B e. A ) -> -. B R B )
Theorem	sotr 4073	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 ) )
Theorem	issod 4074*	An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4052). (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 )
Theorem	sowlin 4075	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 ) ) )
Theorem	so2nr 4076	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 ) )
Theorem	so3nr 4077	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 ) )
Theorem	sotricim 4078	One direction of sotritric 4079 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 ) ) )
Theorem	sotritric 4079	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 ) ) )
Theorem	sotritrieq 4080	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 ) ) )
Theorem	so0 4081	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
Syntax	wfrfor 4082	Extend wff notation to include the well-founded predicate.
wff FrFor R A S
Syntax	wfr 4083	Extend wff notation to include the well-founded predicate. Read: ' R is a well-founded relation on A.'
wff R Fr A
Syntax	wse 4084	Extend wff notation to include the set-like predicate. Read: ' R is set-like on A.'
wff R Se A
Syntax	wwe 4085	Extend wff notation to include the well-ordering predicate. Read: ' R well-orders A.'
wff R We A
Definition	df-frfor 4086*	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 ) )
Definition	df-frind 4087*	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 )
Definition	df-se 4088*	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 )
Definition	df-wetr 4089*	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 4265). 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 ) ) )
Theorem	seex 4090*	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 )
Theorem	exse 4091	Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
|- ( A e. V -> R Se A )
Theorem	sess1 4092	Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( R C_ S -> ( S Se A -> R Se A ) )
Theorem	sess2 4093	Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( A C_ B -> ( R Se B -> R Se A ) )
Theorem	seeq1 4094	Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( R = S -> ( R Se A <-> S Se A ) )
Theorem	seeq2 4095	Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( A = B -> ( R Se A <-> R Se B ) )
Theorem	nfse 4096	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
Theorem	epse 4097	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
Theorem	frforeq1 4098	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
|- ( R = S -> (FrFor R A T <-> FrFor S A T ) )
Theorem	freq1 4099	Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
|- ( R = S -> ( R Fr A <-> S Fr A ) )
Theorem	frforeq2 4100	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
|- ( A = B -> (FrFor R A T <-> FrFor R B T ) )