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

Theoremxpun 4401 The cross product of two unions. (Contributed by NM, 12-Aug-2004.)

Theoremelvv 4402* Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)

Theoremelvvv 4403* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)

Theoremelvvuni 4404 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)

Theoremmosubopt 4405* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)

Theoremmosubop 4406* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)

Theorembrinxp2 4407 Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrinxp 4408 Intersection of binary relation with cross product. (Contributed by NM, 9-Mar-1997.)

Theorempoinxp 4409 Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremsoinxp 4410 Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremseinxp 4411 Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se Se

Theoremposng 4412 Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)

Theoremsosng 4413 Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)

Theoremopabssxp 4414* An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)

Theorembrab2ga 4415* The law of concretion for a binary relation. See brab2a 4393 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)

Theoremoptocl 4416* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)

Theorem2optocl 4417* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theorem3optocl 4418* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theoremopbrop 4419* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)

Theorem0xp 4420 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 4421 Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremreleq 4422 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)

Theoremreleqi 4423 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)

Theoremreleqd 4424 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)

Theoremnfrel 4425 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremsbcrel 4426 Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Theoremrelss 4427 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)

Theoremssrel 4428* 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 4429* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)

Theoremssrel2 4430* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4428 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)

Theoremrelssi 4431* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)

Theoremrelssdv 4432* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)

Theoremeqrelriv 4433* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)

Theoremeqrelriiv 4434* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)

Theoremeqbrriv 4435* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)

Theoremeqrelrdv 4436* Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremeqbrrdv 4437* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremeqbrrdiv 4438* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremeqrelrdv2 4439* A version of eqrelrdv 4436. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremssrelrel 4440* A subclass relationship determined by ordered triples. Use relrelss 4844 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 4441* Extensionality principle for ordered triples, analogous to eqrel 4429. Use relrelss 4844 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)

Theoremelrel 4442* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)

Theoremrelsn 4443 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)

Theoremrelsnop 4444 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremxpss12 4445 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 4446 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrelxp 4447 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)

Theoremxpss1 4448 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpss2 4449 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpsspw 4450 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)

Theoremunixpss 4451 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)

Theoremxpexg 4452 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremxpex 4453 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremrelun 4454 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)

Theoremrelin1 4455 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)

Theoremrelin2 4456 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)

Theoremreldif 4457 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)

Theoremreliun 4458 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)

Theoremreliin 4459 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 4460* 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 4461* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremrel0 4462 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)

Theoremrelopabi 4463 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)

Theoremrelopab 4464 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 4465 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 4466 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremopabid2 4467* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)

Theoreminopab 4468* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Theoremdifopab 4469* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoreminxp 4470 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 4471 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpindir 4472 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpiindim 4473* Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)

Theoremxpriindim 4474* Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)

Theoremeliunxp 4475* Membership in a union of cross products. Analogue of elxp 4362 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremopeliunxp2 4476* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremraliunxp 4477* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4479, is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremrexiunxp 4478* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4480, is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremralxp 4479* Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremrexxp 4480* Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)

Theoremdjussxp 4481* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremralxpf 4482* Version of ralxp 4479 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrexxpf 4483* Version of rexxp 4480 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremiunxpf 4484* Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)

Theoremopabbi2dv 4485* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2156. (Contributed by NM, 24-Feb-2014.)

Theoremrelop 4486* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)

Theoremideqg 4487 For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremideq 4488 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)

Theoremididg 4489 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremissetid 4490 Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremcoss1 4491 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)

Theoremcoss2 4492 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)

Theoremcoeq1 4493 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq2 4494 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq1i 4495 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2i 4496 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq1d 4497 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2d 4498 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq12i 4499 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Theoremcoeq12d 4500 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)

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