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

Theorem0nelelxp 4401 A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)

Theoremopelxp 4402 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 4403 Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)

Theoremopelxpi 4404 Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.)

Theoremopelxp1 4405 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 4406 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 4407 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 4408* Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)

Theorembrrelex12 4409 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.)

Theorembrrelex 4410 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 4411 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.)

Theorembrrelexi 4412 The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)

Theorembrrelex2i 4413 The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremnprrel 4414 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)

Theoremfconstmpt 4415* 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 4416* Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopelvvg 4417 Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)

Theoremopelvv 4418 Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopthprc 4419 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 4420 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 4421* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)

Theoremelxp3 4422* Membership in a cross product. (Contributed by NM, 5-Mar-1995.)

Theoremopeliunxp 4423 Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)

Theoremxpundi 4424 Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)

Theoremxpundir 4425 Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)

Theoremxpiundi 4426* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremxpiundir 4427* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremiunxpconst 4428* Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremxpun 4429 The cross product of two unions. (Contributed by NM, 12-Aug-2004.)

Theoremelvv 4430* Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)

Theoremelvvv 4431* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)

Theoremelvvuni 4432 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)

Theoremmosubopt 4433* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)

Theoremmosubop 4434* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)

Theorembrinxp2 4435 Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrinxp 4436 Intersection of binary relation with cross product. (Contributed by NM, 9-Mar-1997.)

Theorempoinxp 4437 Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremsoinxp 4438 Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremseinxp 4439 Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se Se

Theoremposng 4440 Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)

Theoremsosng 4441 Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)

Theoremopabssxp 4442* An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)

Theorembrab2ga 4443* The law of concretion for a binary relation. See brab2a 4421 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)

Theoremoptocl 4444* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)

Theorem2optocl 4445* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theorem3optocl 4446* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theoremopbrop 4447* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)

Theorem0xp 4448 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 4449 Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremreleq 4450 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)

Theoremreleqi 4451 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)

Theoremreleqd 4452 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)

Theoremnfrel 4453 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremsbcrel 4454 Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Theoremrelss 4455 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)

Theoremssrel 4456* 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 4457* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)

Theoremssrel2 4458* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4456 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)

Theoremrelssi 4459* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)

Theoremrelssdv 4460* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)

Theoremeqrelriv 4461* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)

Theoremeqrelriiv 4462* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)

Theoremeqbrriv 4463* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)

Theoremeqrelrdv 4464* Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremeqbrrdv 4465* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremeqbrrdiv 4466* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremeqrelrdv2 4467* A version of eqrelrdv 4464. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremssrelrel 4468* A subclass relationship determined by ordered triples. Use relrelss 4872 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 4469* Extensionality principle for ordered triples, analogous to eqrel 4457. Use relrelss 4872 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)

Theoremelrel 4470* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)

Theoremrelsn 4471 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)

Theoremrelsnop 4472 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremxpss12 4473 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 4474 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrelxp 4475 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)

Theoremxpss1 4476 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpss2 4477 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpsspw 4478 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)

Theoremunixpss 4479 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)

Theoremxpexg 4480 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremxpex 4481 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremrelun 4482 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)

Theoremrelin1 4483 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)

Theoremrelin2 4484 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)

Theoremreldif 4485 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)

Theoremreliun 4486 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)

Theoremreliin 4487 An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremreluni 4488* The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)

Theoremrelint 4489* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremrel0 4490 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)

Theoremrelopabi 4491 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)

Theoremrelopab 4492 A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)

Theoremreli 4493 The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremrele 4494 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremopabid2 4495* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)

Theoreminopab 4496* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Theoremdifopab 4497* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoreminxp 4498 The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremxpindi 4499 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpindir 4500 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

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