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Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxpeq12i 4601 Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)

Theoremxpeq1d 4602 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremxpeq2d 4603 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremxpeq12d 4604 Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)

Theoremsqxpeqd 4605 Equality deduction for a Cartesian square, see Wikipedia "Cartesian product", https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power. (Contributed by AV, 13-Jan-2020.)

Theoremnfxp 4606 Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theorem0nelxp 4607 The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorem0nelelxp 4608 A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)

Theoremopelxp 4609 Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrxp 4610 Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)

Theoremopelxpi 4611 Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.)

Theoremopelxpd 4612 Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Theoremopelxp1 4613 The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopelxp2 4614 The second member of an ordered pair of classes in a cross product belongs to second cross product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremotelxp1 4615 The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)

Theoremrabxp 4616* Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)

Theorembrrelex12 4617 A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorembrrelex1 4618 A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrrelex 4619 A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrrelex2 4620 A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorembrrelex12i 4621 Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.)

Theorembrrelex1i 4622 The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)

Theorembrrelex2i 4623 The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremnprrel 4624 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)

Theorem0nelrel 4625 A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)

Theoremfconstmpt 4626* Representation of a constant function using the mapping operation. (Note that cannot appear free in .) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremvtoclr 4627* Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopelvvg 4628 Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)

Theoremopelvv 4629 Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopthprc 4630 Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)

Theorembrel 4631 Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrab2a 4632* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)

Theoremelxp3 4633* Membership in a cross product. (Contributed by NM, 5-Mar-1995.)

Theoremopeliunxp 4634 Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)

Theoremxpundi 4635 Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)

Theoremxpundir 4636 Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)

Theoremxpiundi 4637* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremxpiundir 4638* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremiunxpconst 4639* Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremxpun 4640 The cross product of two unions. (Contributed by NM, 12-Aug-2004.)

Theoremelvv 4641* Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)

Theoremelvvv 4642* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)

Theoremelvvuni 4643 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)

Theoremmosubopt 4644* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)

Theoremmosubop 4645* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)

Theorembrinxp2 4646 Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrinxp 4647 Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)

Theorempoinxp 4648 Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremsoinxp 4649 Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremseinxp 4650 Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se Se

Theoremposng 4651 Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)

Theoremsosng 4652 Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)

Theoremopabssxp 4653* An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)

Theorembrab2ga 4654* The law of concretion for a binary relation. See brab2a 4632 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)

Theoremoptocl 4655* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)

Theorem2optocl 4656* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theorem3optocl 4657* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theoremopbrop 4658* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)

Theorem0xp 4659 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)

Theoremcsbxpg 4660 Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremreleq 4661 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)

Theoremreleqi 4662 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)

Theoremreleqd 4663 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)

Theoremnfrel 4664 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremsbcrel 4665 Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Theoremrelss 4666 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)

Theoremssrel 4667* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremeqrel 4668* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)

Theoremssrel2 4669* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4667 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)

Theoremrelssi 4670* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)

Theoremrelssdv 4671* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)

Theoremeqrelriv 4672* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)

Theoremeqrelriiv 4673* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)

Theoremeqbrriv 4674* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)

Theoremeqrelrdv 4675* Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremeqbrrdv 4676* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremeqbrrdiv 4677* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremeqrelrdv2 4678* A version of eqrelrdv 4675. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremssrelrel 4679* A subclass relationship determined by ordered triples. Use relrelss 5105 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremeqrelrel 4680* Extensionality principle for ordered triples, analogous to eqrel 4668. Use relrelss 5105 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)

Theoremelrel 4681* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)

Theoremrelsng 4682 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)

Theoremrelsnopg 4683 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by BJ, 12-Feb-2022.)

Theoremrelsn 4684 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)

Theoremrelsnop 4685 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremxpss12 4686 Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremxpss 4687 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrelxp 4688 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)

Theoremxpss1 4689 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpss2 4690 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpsspw 4691 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)

Theoremunixpss 4692 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)

Theoremxpexg 4693 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremxpex 4694 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremsqxpexg 4695 The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.)

Theoremrelun 4696 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)

Theoremrelin1 4697 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)

Theoremrelin2 4698 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)

Theoremreldif 4699 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)

Theoremreliun 4700 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)

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