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Theorem List for Intuitionistic Logic Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsoirri 4901 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri 4902 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremson2lpi 4903 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri2 4904 A transitivity relation. (Read B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

Theoremsotri3 4905 A transitivity relation. (Read A < B and C < B implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

Theorempoleloe 4906 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theorempoltletr 4907 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremcnvopab 4908* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremmptcnv 4909* The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)

Theoremcnv0 4910 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)

Theoremcnvi 4911 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 4912 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 4913 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremcnvin 4914 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 4915 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Theoremrnin 4916 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremrniun 4917 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremrnuni 4918* 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 4919 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremimaundir 4920 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Theoremdminss 4921 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 4922 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Theoreminimass 4923 The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoreminimasn 4924 The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremcnvxp 4925 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 4926 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 4927* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)

Theoremxpm 4928* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)

Theoremxpeq0r 4929 A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremsqxpeq0 4930 A Cartesian square is empty iff its member is empty. (Contributed by Jim Kingdon, 21-Apr-2023.)

Theoremxpdisj1 4931 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpdisj2 4932 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpsndisj 4933 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)

Theoremdjudisj 4934* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremresdisj 4935 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 4936* The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with nonempty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremdmxpss 4937 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)

Theoremrnxpss 4938 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.)

Theoremdmxpss2 4939 Upper bound for the domain of a binary relation. (Contributed by BJ, 10-Jul-2022.)

Theoremrnxpss2 4940 Upper bound for the range of a binary relation. (Contributed by BJ, 10-Jul-2022.)

Theoremrnxpid 4941 The range of a square cross product. (Contributed by FL, 17-May-2010.)

Theoremssxpbm 4942* A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremssxp1 4943* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremssxp2 4944* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremxp11m 4945* The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)

Theoremxpcanm 4946* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremxpcan2m 4947* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremxpexr2m 4948* If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremssrnres 4949 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)

Theoremrninxp 4950* Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdminxp 4951* Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)

Theoremimainrect 4952 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremxpima1 4953 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremxpima2m 4954* The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremxpimasn 4955 The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Theoremcnvcnv3 4956* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremdfrel2 4957 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)

Theoremdfrel4v 4958* A relation can be expressed as the set of ordered pairs in it. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremcnvcnv 4959 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)

Theoremcnvcnv2 4960 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)

Theoremcnvcnvss 4961 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)

Theoremcnveqb 4962 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)

Theoremcnveq0 4963 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)

Theoremdfrel3 4964 Alternate definition of relation. (Contributed by NM, 14-May-2008.)

Theoremdmresv 4965 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremrnresv 4966 The range of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremdfrn4 4967 Range defined in terms of image. (Contributed by NM, 14-May-2008.)

Theoremcsbrng 4968 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremrescnvcnv 4969 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvcnvres 4970 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)

Theoremimacnvcnv 4971 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Theoremdmsnm 4972* The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)

Theoremrnsnm 4973* The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)

Theoremdmsn0 4974 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)

Theoremcnvsn0 4975 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremdmsn0el 4976 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)

Theoremrelsn2m 4977* A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)

Theoremdmsnopg 4978 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmpropg 4979 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnop 4980 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 4981 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)

Theoremdmtpop 4982 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)

Theoremcnvcnvsn 4983 Double converse of a singleton of an ordered pair. (Unlike cnvsn 4989, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnsnsng 4984 The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)

Theoremrnsnopg 4985 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 4986 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremrnsnop 4987 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, 26-Apr-2015.)

Theoremop1sta 4988 Extract the first member of an ordered pair. (See op2nda 4991 to extract the second member and op1stb 4367 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)

Theoremcnvsn 4989 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremop2ndb 4990 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4367 to extract the first member and op2nda 4991 for an alternate version.) (Contributed by NM, 25-Nov-2003.)

Theoremop2nda 4991 Extract the second member of an ordered pair. (See op1sta 4988 to extract the first member and op2ndb 4990 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvsng 4992 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)

Theoremopswapg 4993 Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)

Theoremelxp4 4994 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4995. (Contributed by NM, 17-Feb-2004.)

Theoremelxp5 4995 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4994 when the double intersection does not create class existence problems (caused by int0 3753). (Contributed by NM, 1-Aug-2004.)

Theoremcnvresima 4996 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Theoremresdm2 4997 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)

Theoremresdmres 4998 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremimadmres 4999 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremmptpreima 5000* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

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