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Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfiu1 3901 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
𝑥 𝑥𝐴 𝐵
 
Theoremnfii1 3902 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
𝑥 𝑥𝐴 𝐵
 
Theoremdfiun2g 3903* Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
 
Theoremdfiin2g 3904* Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
 
Theoremdfiun2 3905* Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
𝐵 ∈ V        𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
 
Theoremdfiin2 3906* Alternate definition of indexed intersection when 𝐵 is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝐵 ∈ V        𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
 
Theoremdfiunv2 3907* Define double indexed union. (Contributed by FL, 6-Nov-2013.)
𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}
 
Theoremcbviun 3908* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
Theoremcbviin 3909* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
Theoremcbviunv 3910* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
Theoremcbviinv 3911* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
Theoremiunss 3912* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
 
Theoremssiun 3913* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∃𝑥𝐴 𝐶𝐵𝐶 𝑥𝐴 𝐵)
 
Theoremssiun2 3914 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝑥𝐴𝐵 𝑥𝐴 𝐵)
 
Theoremssiun2s 3915* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
(𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)
 
Theoremiunss2 3916* A subclass condition on the members of two indexed classes 𝐶(𝑥) and 𝐷(𝑦) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3825. (Contributed by NM, 9-Dec-2004.)
(∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
 
Theoremiunab 3917* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
 
Theoremiunrab 3918* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}
 
Theoremiunxdif2 3919* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
(𝑥 = 𝑦𝐶 = 𝐷)       (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)
 
Theoremssiinf 3920 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
𝑥𝐶       (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 
Theoremssiin 3921* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
(𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 
Theoremiinss 3922* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
 
Theoremiinss2 3923 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
(𝑥𝐴 𝑥𝐴 𝐵𝐵)
 
Theoremuniiun 3924* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥
 
Theoremintiin 3925* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥
 
Theoremiunid 3926* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
𝑥𝐴 {𝑥} = 𝐴
 
Theoremiun0 3927 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥𝐴 ∅ = ∅
 
Theorem0iun 3928 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥 ∈ ∅ 𝐴 = ∅
 
Theorem0iin 3929 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝐴 = V
 
Theoremviin 3930* Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
 
Theoremiunn0m 3931* There is an inhabited class in an indexed collection 𝐵(𝑥) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)
(∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
 
Theoremiinab 3932* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}
 
Theoremiinrabm 3933* Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
(∃𝑥 𝑥𝐴 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
 
Theoremiunin2 3934* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3924 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
 
Theoremiunin1 3935* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3924 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
 
Theoremiundif2ss 3936* Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
𝑥𝐴 (𝐵𝐶) ⊆ (𝐵 𝑥𝐴 𝐶)
 
Theorem2iunin 3937* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
 
Theoremiindif2m 3938* Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
 
Theoremiinin2m 3939* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
 
Theoremiinin1m 3940* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥 𝑥𝐴 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
 
Theoremelriin 3941* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
 
Theoremriin0 3942* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
 
Theoremriinm 3943* Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
 
Theoremiinxsng 3944* A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiinxprg 3945* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
 
Theoremiunxsng 3946* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiunxsn 3947* A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ {𝐴}𝐵 = 𝐶
 
Theoremiunxsngf 3948* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiunun 3949 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
 
Theoremiunxun 3950 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
 
Theoremiunxprg 3951* A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
 
Theoremiunxiun 3952* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
 
Theoremiinuniss 3953* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
(𝐴 𝐵) ⊆ 𝑥𝐵 (𝐴𝑥)
 
Theoremiununir 3954* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)
((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
 
Theoremsspwuni 3955 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
 
Theorempwssb 3956* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremelpwpw 3957 Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
 
Theorempwpwab 3958* The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)
𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}
 
Theorempwpwssunieq 3959* The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
{𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
 
Theoremelpwuni 3960 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))
 
Theoremiinpw 3961* The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
 
Theoremiunpwss 3962* Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
 
Theoremrintm 3963* Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑋) = 𝑋)
 
2.1.21  Disjointness
 
Syntaxwdisj 3964 Extend wff notation to include the statement that a family of classes 𝐵(𝑥), for 𝑥𝐴, is a disjoint family.
wff Disj 𝑥𝐴 𝐵
 
Definitiondf-disj 3965* A collection of classes 𝐵(𝑥) is disjoint when for each element 𝑦, it is in 𝐵(𝑥) for at most one 𝑥. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
 
Theoremdfdisj2 3966* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
 
Theoremdisjss2 3967 If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
 
Theoremdisjeq2 3968 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq2dv 3969* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
 
Theoremdisjss1 3970* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq1 3971* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjeq1d 3972* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjeq12d 3973* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
 
Theoremcbvdisj 3974* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremcbvdisjv 3975* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
(𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremnfdisjv 3976* Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
𝑦𝐴    &   𝑦𝐵       𝑦Disj 𝑥𝐴 𝐵
 
Theoremnfdisj1 3977 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥Disj 𝑥𝐴 𝐵
 
Theoremdisjnim 3978* If a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)
(𝑖 = 𝑗𝐵 = 𝐶)       (Disj 𝑖𝐴 𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝐵𝐶) = ∅))
 
Theoremdisjnims 3979* If a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)
(Disj 𝑥𝐴 𝐵 → ∀𝑖𝐴𝑗𝐴 (𝑖𝑗 → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
 
Theoremdisji2 3980* Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑋𝑌, then 𝐶 and 𝐷 are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
 
Theoreminvdisj 3981* If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥𝐴 are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
(∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥Disj 𝑥𝐴 𝐵)
 
Theoremdisjiun 3982* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
((Disj 𝑥𝐴 𝐵 ∧ (𝐶𝐴𝐷𝐴 ∧ (𝐶𝐷) = ∅)) → ( 𝑥𝐶 𝐵 𝑥𝐷 𝐵) = ∅)
 
Theoremsndisj 3983 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴 {𝑥}
 
Theorem0disj 3984 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴
 
Theoremdisjxsn 3985* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ {𝐴}𝐵
 
Theoremdisjx0 3986 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ ∅ 𝐵
 
2.1.22  Binary relations
 
Syntaxwbr 3987 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous.
wff 𝐴𝑅𝐵
 
Definitiondf-br 3988 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. This definition of relations is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when 𝑅 is a proper class (see for example iprc 4877). (Contributed by NM, 31-Dec-1993.)
(𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
 
Theorembreq 3989 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
(𝑅 = 𝑆 → (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembreq1 3990 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreq2 3991 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12 3992 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqi 3993 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
𝑅 = 𝑆       (𝐴𝑅𝐵𝐴𝑆𝐵)
 
Theorembreq1i 3994 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐴𝑅𝐶𝐵𝑅𝐶)
 
Theorembreq2i 3995 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐶𝑅𝐴𝐶𝑅𝐵)
 
Theorembreq12i 3996 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝑅𝐶𝐵𝑅𝐷)
 
Theorembreq1d 3997 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreqd 3998 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
 
Theorembreq2d 3999 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12d 4000 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
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