P. 44 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4301-4400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rext 4301*	A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
|- ( A. z ( x e. z -> y e. z ) -> x = y )
Theorem	sspwb 4302	Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
|- ( A C_ B <-> ~P A C_ ~P B )
Theorem	unipw 4303	A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
|- U. ~P A = A
Theorem	pwel 4304	Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
|- ( A e. B -> ~P A e. ~P ~P U. B )
Theorem	pwtr 4305	A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
|- ( Tr A <-> Tr ~P A )
Theorem	ssextss 4306*	An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
|- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )
Theorem	ssext 4307*	An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
|- ( A = B <-> A. x ( x C_ A <-> x C_ B ) )
Theorem	nssssr 4308*	Negation of subclass relationship. Compare nssr 3284. (Contributed by Jim Kingdon, 17-Sep-2018.)
|- ( E. x ( x C_ A /\ -. x C_ B ) -> -. A C_ B )
Theorem	pweqb 4309	Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
|- ( A = B <-> ~P A = ~P B )
Theorem	intid 4310*	The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
|- A e. _V   =>    |- |^| { x | A e. x } = { A }
Theorem	euabex 4311	The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
|- ( E! x ph -> { x | ph } e. _V )
Theorem	mss 4312*	An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
|- ( E. y y e. A -> E. x ( x C_ A /\ E. z z e. x ) )
Theorem	exss 4313*	Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
|- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) )
Theorem	opexg 4314	An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.)
|- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
Theorem	opex 4315	An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.)
|- A e. _V   &    |- B e. _V   =>    |- <. A , B >. e. _V
Theorem	otexg 4316	An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.)
|- ( ( A e. U /\ B e. V /\ C e. W ) -> <. A , B , C >. e. _V )
Theorem	elop 4317	An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   =>    |- ( A e. <. B , C >. <-> ( A = { B } \/ A = { B , C } ) )
Theorem	opi1 4318	One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- { A } e. <. A , B >.
Theorem	opi2 4319	One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- { A , B } e. <. A , B >.
2.3.4  Ordered pair theorem
Theorem	opm 4320*	An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)
|- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
Theorem	opnzi 4321	An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4320). (Contributed by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- <. A , B >. =/= (/)
Theorem	opth1 4322	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 )
Theorem	opth 4323	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 ) )
Theorem	opthg 4324	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 ) ) )
Theorem	opthg2 4325	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 ) ) )
Theorem	opth2 4326	Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
|- C e. _V   &    |- D e. _V   =>    |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )
Theorem	otth2 4327	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 ) )
Theorem	otth 4328	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 ) )
Theorem	eqvinop 4329*	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 >. ) )
Theorem	copsexg 4330*	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 ) ) )
Theorem	copsex2t 4331*	Closed theorem form of copsex2g 4332. (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 ) )
Theorem	copsex2g 4332*	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 4333*	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 4334	A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- -. (/) e. <. A , B >.
Theorem	opwo0id 4335	An ordered pair is equal to the ordered pair without the empty set. This is because no ordered pair contains the empty set. (Contributed by AV, 15-Nov-2021.)
|- <. X , Y >. = ( <. X , Y >. \ { (/) } )
Theorem	opeqex 4336	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 4337	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 4338*	"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 4339	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 4340	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 4341*	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 4342	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 4343	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.)
Theorem	opabid 4344	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	opabidw 4345*	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4344 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 4346*	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 4347*	The law of concretion in terms of substitutions. Less general than opelopabsb 4348, 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 4348*	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 4349*	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 4350*	Closed theorem form of opelopab 4360. (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 4351*	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 4352*	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 4353*	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 4354*	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 4355*	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 4356*	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 4357*	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 4358*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4356 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 4359*	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 4360*	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 4361*	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 4362*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4360 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 4363*	The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4360 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 4364	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 4365	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 4366	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 4367*	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 4368	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 4369*	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 4370*	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 4371*	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 4372*	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 4373	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 4374	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 4375	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 4376	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 4377	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 4378	Extend class notation to include the epsilon relation.
class _E
Syntax	cid 4379	Extend the definition of a class to include identity relation.
class _I
Definition	df-eprel 4380*	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 4381. Thus, 5 _E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.)
|- _E = { <. x , y >. | x e. y }
Theorem	epelg 4381	The epsilon relation and membership are the same. General version of epel 4383. (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 4382	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 4383	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 4384*	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 4726), 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 4385	Extend wff notation to include the strict partial ordering predicate. Read: ' R is a partial order on A.'
wff R Po A
Syntax	wor 4386	Extend wff notation to include the strict linear ordering predicate. Read: ' R orders A.'
wff R Or A
Definition	df-po 4387*	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 4388*	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 4389	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 4390	Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
|- ( R = S -> ( R Po A <-> S Po A ) )
Theorem	poeq2 4391	Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
|- ( A = B -> ( R Po A <-> R Po B ) )
Theorem	nfpo 4392	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 4393	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 4394	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 4395*	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 4396*	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 4397*	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 4398	A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
|- ( ( R Po A /\ B e. A ) -> -. B R B )
Theorem	potr 4399	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 4400	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 ) )