P. 45 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opabidw 4401*	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4400 with a disjoint variable condition. (Contributed by NM, 14-Apr-1995.) (Revised by GG, 26-Jan-2024.)
|- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
Theorem	elopab 4402*	Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.)
|- ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) )
Theorem	opelopabsbALT 4403*	The law of concretion in terms of substitutions. Less general than opelopabsb 4404, 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 4404*	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 4405*	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 4406*	Closed theorem form of opelopab 4416. (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 4407*	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 4408*	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 4409*	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 4410*	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 4411*	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 4412*	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 4413*	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	opelopabgf 4414*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4412 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
|- F/ x ps   &    |- F/ y ch   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )
Theorem	opelopab2 4415*	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 4416*	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 4417*	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 4418*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4416 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 4419*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4416 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 4420	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 4421	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 4422	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 4423*	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 4424	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 4425*	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 4426*	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 }
Theorem	elopabr 4427*	Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.)
|- ( A e. { <. x , y >. | x R y } -> A e. R )
Theorem	elopabran 4428*	Membership in an ordered-pair class abstraction defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021.)
|- ( A e. { <. x , y >. | ( x R y /\ ps ) } -> A e. R )
2.3.6  Power class of union and intersection
Theorem	pwin 4429	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 4430	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 4431	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 4432	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 4433	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
Syntax	cep 4434	Extend class notation to include the epsilon relation.
class _E
Syntax	cid 4435	Extend the definition of a class to include identity relation.
class _I
Definition	df-eprel 4436*	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 4437. Thus, 5 _E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)
|- _E = { <. x , y >. | x e. y }
Theorem	epelg 4437	The epsilon relation and membership are the same. General version of epel 4439. (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 4438	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 4439	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 4440*	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 total orderings We have not yet defined relations (df-rel 4783), but here we introduce a few related notions we will use to develop ordinals. The class variable R is no different from other class variables, but it reminds us that typically it represents what we will later call a "relation".
Syntax	wpo 4441	Extend wff notation to include the strict partial ordering predicate. Read: ' R is a partial order on A.'
wff R Po A
Syntax	wor 4442	Extend wff notation to include the strict linear ordering predicate. Read: ' R orders A.'
wff R Or A
Definition	df-po 4443*	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 4444*	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 4445	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 4446	Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
|- ( R = S -> ( R Po A <-> S Po A ) )
Theorem	poeq2 4447	Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
|- ( A = B -> ( R Po A <-> R Po B ) )
Theorem	nfpo 4448	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 4449	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 4450	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 4451*	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 4452*	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 4453*	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 4454	A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
|- ( ( R Po A /\ B e. A ) -> -. B R B )
Theorem	potr 4455	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 4456	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 4457	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 4458	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 4459*	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 4460	A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
|- ( R Or A -> R Po A )
Theorem	soss 4461	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 4462	Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
|- ( R = S -> ( R Or A <-> S Or A ) )
Theorem	soeq2 4463	Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
|- ( A = B -> ( R Or A <-> R Or B ) )
Theorem	sonr 4464	A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
|- ( ( R Or A /\ B e. A ) -> -. B R B )
Theorem	sotr 4465	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 4466*	An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4444). (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 4467	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 4468	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 4469	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 4470	One direction of sotritric 4471 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 4471	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 4472	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 4473	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
Syntax	wfrfor 4474	Extend wff notation to include the well-founded predicate.
wff FrFor R A S
Syntax	wfr 4475	Extend wff notation to include the well-founded predicate. Read: ' R is a well-founded relation on A.'
wff R Fr A
Syntax	wse 4476	Extend wff notation to include the set-like predicate. Read: ' R is set-like on A.'
wff R Se A
Syntax	wwe 4477	Extend wff notation to include the well-ordering predicate. Read: ' R well-orders A.'
wff R We A
Definition	df-frfor 4478*	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 4479*	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 4480*	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 4481*	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 do not have that as seen at ordtriexmid 4670). Given excluded middle, well-ordering is usually defined to require trichotomy (and the definition 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 4482*	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 4483	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 4484	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 4485	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 4486	Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( R = S -> ( R Se A <-> S Se A ) )
Theorem	seeq2 4487	Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( A = B -> ( R Se A <-> R Se B ) )
Theorem	nfse 4488	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 4489	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 4490	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 4491	Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
|- ( R = S -> ( R Fr A <-> S Fr A ) )
Theorem	frforeq2 4492	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
|- ( A = B -> (FrFor R A T <-> FrFor R B T ) )
Theorem	freq2 4493	Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
|- ( A = B -> ( R Fr A <-> R Fr B ) )
Theorem	frforeq3 4494	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
|- ( S = T -> (FrFor R A S <-> FrFor R A T ) )
Theorem	nffrfor 4495	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
Theorem	nffr 4496	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
Theorem	frirrg 4497	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 )
Theorem	fr0 4498	Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
|- R Fr (/)
Theorem	frind 4499*	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 )
Theorem	efrirr 4500	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 )