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Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremimass2 4701 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)

Theoremndmima 4702 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)

Theoremrelcnv 4703 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)

Theoremrelbrcnvg 4704 When is a relation, the sethood assumptions on brcnv 4518 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremrelbrcnv 4705 When is a relation, the sethood assumptions on brcnv 4518 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremcotr 4706* Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremissref 4707* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)

Theoremcnvsym 4708* Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremintasym 4709* Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremasymref 4710* Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 4846. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremintirr 4711* Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theorembrcodir 4712* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)

Theoremcodir 4713* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremqfto 4714* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremxpidtr 4715 A square cross product is a transitive relation. (Contributed by FL, 31-Jul-2009.)

Theoremtrin2 4716 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)

Theorempoirr2 4717 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)

Theoremtrinxp 4718 The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square cross product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)

Theoremsoirri 4719 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri 4720 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremson2lpi 4721 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri2 4722 A transitivity relation. (Read B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

Theoremsotri3 4723 A transitivity relation. (Read A < B and C < B implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

Theorempoleloe 4724 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theorempoltletr 4725 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremcnvopab 4726* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnv0 4727 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)

Theoremcnvi 4728 The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvun 4729 The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvdif 4730 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremcnvin 4731 Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremrnun 4732 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Theoremrnin 4733 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremrniun 4734 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremrnuni 4735* The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.)

Theoremimaundi 4736 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremimaundir 4737 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Theoremdminss 4738 An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.)

Theoremimainss 4739 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Theoreminimass 4740 The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoreminimasn 4741 The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremcnvxp 4742 The converse of a cross product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremxp0 4743 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)

Theoremxpmlem 4744* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)

Theoremxpm 4745* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)

Theoremxpeq0r 4746 A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremxpdisj1 4747 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpdisj2 4748 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpsndisj 4749 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)

Theoremdjudisj 4750* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremresdisj 4751 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnxpm 4752* The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremdmxpss 4753 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)

Theoremrnxpss 4754 The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnxpid 4755 The range of a square cross product. (Contributed by FL, 17-May-2010.)

Theoremssxpbm 4756* A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremssxp1 4757* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremssxp2 4758* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremxp11m 4759* The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)

Theoremxpcanm 4760* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremxpcan2m 4761* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremxpexr2m 4762* If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremssrnres 4763 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)

Theoremrninxp 4764* Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdminxp 4765* Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)

Theoremimainrect 4766 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremxpima1 4767 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremxpima2m 4768* The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremxpimasn 4769 The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Theoremcnvcnv3 4770* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremdfrel2 4771 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)

Theoremdfrel4v 4772* A relation can be expressed as the set of ordered pairs in it. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremcnvcnv 4773 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)

Theoremcnvcnv2 4774 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)

Theoremcnvcnvss 4775 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)

Theoremcnveqb 4776 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)

Theoremcnveq0 4777 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)

Theoremdfrel3 4778 Alternate definition of relation. (Contributed by NM, 14-May-2008.)

Theoremdmresv 4779 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremrnresv 4780 The range of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremdfrn4 4781 Range defined in terms of image. (Contributed by NM, 14-May-2008.)

Theoremcsbrng 4782 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremrescnvcnv 4783 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvcnvres 4784 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)

Theoremimacnvcnv 4785 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Theoremdmsnm 4786* The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)

Theoremrnsnm 4787* The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)

Theoremdmsn0 4788 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)

Theoremcnvsn0 4789 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremdmsn0el 4790 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)

Theoremrelsn2m 4791* A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)

Theoremdmsnopg 4792 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmpropg 4793 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnop 4794 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremdmprop 4795 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)

Theoremdmtpop 4796 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)

Theoremcnvcnvsn 4797 Double converse of a singleton of an ordered pair. (Unlike cnvsn 4803, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnsnsng 4798 The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)

Theoremrnsnopg 4799 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrnpropg 4800 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)

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