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Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuniiun 3701* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
A = x A x

Theoremintiin 3702* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
A = x A x

Theoremiunid 3703* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
x A {x} = A

Theoremiun0 3704 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
x A ∅ = ∅

Theorem0iun 3705 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
x A = ∅

Theorem0iin 3706 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
x A = V

Theoremviin 3707* Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
x V A = {yx y A}

Theoremiunn0m 3708* 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.)
(x A y y By y x A B)

Theoremiinab 3709* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
x A {yφ} = {yx A φ}

Theoremiinrabm 3710* Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
(x x A x A {y Bφ} = {y Bx A φ})

Theoremiunin2 3711* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3701 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
x A (B𝐶) = (B x A 𝐶)

Theoremiunin1 3712* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3701 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
x A (𝐶B) = ( x A 𝐶B)

Theoremiundif2ss 3713* Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
x A (B𝐶) ⊆ (B x A 𝐶)

Theorem2iunin 3714* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
x A y B (𝐶𝐷) = ( x A 𝐶 y B 𝐷)

Theoremiindif2m 3715* Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
(x x A x A (B𝐶) = (B x A 𝐶))

Theoremiinin2m 3716* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
(x x A x A (B𝐶) = (B x A 𝐶))

Theoremiinin1m 3717* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
(x x A x A (𝐶B) = ( x A 𝐶B))

Theoremelriin 3718* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
(B (A x 𝑋 𝑆) ↔ (B A x 𝑋 B 𝑆))

Theoremriin0 3719* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (A x 𝑋 𝑆) = A)

Theoremriinm 3720* Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
((x 𝑋 𝑆A x x 𝑋) → (A x 𝑋 𝑆) = x 𝑋 𝑆)

Theoremiinxsng 3721* 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)       (A 𝑉 x {A}B = 𝐶)

Theoremiinxprg 3722* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
(x = A𝐶 = 𝐷)    &   (x = B𝐶 = 𝐸)       ((A 𝑉 B 𝑊) → x {A, B}𝐶 = (𝐷𝐸))

Theoremiunxsng 3723* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
(x = AB = 𝐶)       (A 𝑉 x {A}B = 𝐶)

Theoremiunxsn 3724* 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

Theoremiunun 3725 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
x A (B𝐶) = ( x A B x A 𝐶)

Theoremiunxun 3726 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 = ( x A 𝐶 x B 𝐶)

Theoremiunxiun 3727* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
x y A B𝐶 = y A x B 𝐶

Theoremiinuniss 3728* 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) ⊆ x B (Ax)

Theoremiununir 3729* 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) = x B (Ax) → (B = ∅ → A = ∅))

Theoremsspwuni 3730 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(A ⊆ 𝒫 B AB)

Theorempwssb 3731* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
(A ⊆ 𝒫 Bx A xB)

Theoremelpwuni 3732 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(B A → (A ⊆ 𝒫 B A = B))

Theoremiinpw 3733* 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 = x A 𝒫 x

Theoremiunpwss 3734* 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.)
x A 𝒫 x ⊆ 𝒫 A

Theoremrintm 3735* Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
((𝑋 ⊆ 𝒫 A x x 𝑋) → (A 𝑋) = 𝑋)

2.1.21  Disjointness

Syntaxwdisj 3736 Extend wff notation to include the statement that a family of classes B(x), for x A, is a disjoint family.
wff Disj x A B

Definitiondf-disj 3737* 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 x A By∃*x A y B)

Theoremdfdisj2 3738* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
(Disj x A By∃*x(x A y B))

Theoremdisjss2 3739 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.)
(x A B𝐶 → (Disj x A 𝐶Disj x A B))

Theoremdisjeq2 3740 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(x A B = 𝐶 → (Disj x A BDisj x A 𝐶))

Theoremdisjeq2dv 3741* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
((φ x A) → B = 𝐶)       (φ → (Disj x A BDisj x A 𝐶))

Theoremdisjss1 3742* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(AB → (Disj x B 𝐶Disj x A 𝐶))

Theoremdisjeq1 3743* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(A = B → (Disj x A 𝐶Disj x B 𝐶))

Theoremdisjeq1d 3744* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(φA = B)       (φ → (Disj x A 𝐶Disj x B 𝐶))

Theoremdisjeq12d 3745* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (Disj x A 𝐶Disj x B 𝐷))

Theoremcbvdisj 3746* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
yB    &   x𝐶    &   (x = yB = 𝐶)       (Disj x A BDisj y A 𝐶)

Theoremcbvdisjv 3747* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
(x = yB = 𝐶)       (Disj x A BDisj y A 𝐶)

Theoremnfdisjv 3748* Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
yA    &   yB       yDisj x A B

Theoremnfdisj1 3749 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
xDisj x A B

Theoreminvdisj 3750* If there is a function 𝐶(y) such that 𝐶(y) = x for all y B(x), then the sets B(x) for distinct x A are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
(x A y B 𝐶 = xDisj x A B)

Theoremsndisj 3751 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x A {x}

Theorem0disj 3752 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x A

Theoremdisjxsn 3753* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x {A}B

Theoremdisjx0 3754 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x B

2.1.22  Binary relations

Syntaxwbr 3755 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 3756 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 4543). (Contributed by NM, 31-Dec-1993.)
(A𝑅B ↔ ⟨A, B 𝑅)

Theorembreq 3757 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
(𝑅 = 𝑆 → (A𝑅BA𝑆B))

Theorembreq1 3758 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(A = B → (A𝑅𝐶B𝑅𝐶))

Theorembreq2 3759 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(A = B → (𝐶𝑅A𝐶𝑅B))

Theorembreq12 3760 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
((A = B 𝐶 = 𝐷) → (A𝑅𝐶B𝑅𝐷))

Theorembreqi 3761 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
𝑅 = 𝑆       (A𝑅BA𝑆B)

Theorembreq1i 3762 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
A = B       (A𝑅𝐶B𝑅𝐶)

Theorembreq2i 3763 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
A = B       (𝐶𝑅A𝐶𝑅B)

Theorembreq12i 3764 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 3765 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)       (φ → (A𝑅𝐶B𝑅𝐶))

Theorembreqd 3766 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(φA = B)       (φ → (𝐶A𝐷𝐶B𝐷))

Theorembreq2d 3767 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)       (φ → (𝐶𝑅A𝐶𝑅B))

Theorembreq12d 3768 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))

Theorembreq123d 3769 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(φA = B)    &   (φ𝑅 = 𝑆)    &   (φ𝐶 = 𝐷)       (φ → (A𝑅𝐶B𝑆𝐷))

Theorembreqan12d 3770 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)    &   (ψ𝐶 = 𝐷)       ((φ ψ) → (A𝑅𝐶B𝑅𝐷))

Theorembreqan12rd 3771 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)    &   (ψ𝐶 = 𝐷)       ((ψ φ) → (A𝑅𝐶B𝑅𝐷))

Theoremnbrne1 3772 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((A𝑅B ¬ A𝑅𝐶) → B𝐶)

Theoremnbrne2 3773 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((A𝑅𝐶 ¬ B𝑅𝐶) → AB)

Theoremeqbrtri 3774 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   B𝑅𝐶       A𝑅𝐶

Theoremeqbrtrd 3775 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
(φA = B)    &   (φB𝑅𝐶)       (φA𝑅𝐶)

Theoremeqbrtrri 3776 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   A𝑅𝐶       B𝑅𝐶

Theoremeqbrtrrd 3777 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(φA = B)    &   (φA𝑅𝐶)       (φB𝑅𝐶)

Theorembreqtri 3778 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A𝑅B    &   B = 𝐶       A𝑅𝐶

Theorembreqtrd 3779 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(φA𝑅B)    &   (φB = 𝐶)       (φA𝑅𝐶)

Theorembreqtrri 3780 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A𝑅B    &   𝐶 = B       A𝑅𝐶

Theorembreqtrrd 3781 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(φA𝑅B)    &   (φ𝐶 = B)       (φA𝑅𝐶)

Theorem3brtr3i 3782 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
A𝑅B    &   A = 𝐶    &   B = 𝐷       𝐶𝑅𝐷

Theorem3brtr4i 3783 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
A𝑅B    &   𝐶 = A    &   𝐷 = B       𝐶𝑅𝐷

Theorem3brtr3d 3784 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
(φA𝑅B)    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐶𝑅𝐷)

Theorem3brtr4d 3785 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
(φA𝑅B)    &   (φ𝐶 = A)    &   (φ𝐷 = B)       (φ𝐶𝑅𝐷)

Theorem3brtr3g 3786 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(φA𝑅B)    &   A = 𝐶    &   B = 𝐷       (φ𝐶𝑅𝐷)

Theorem3brtr4g 3787 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(φA𝑅B)    &   𝐶 = A    &   𝐷 = B       (φ𝐶𝑅𝐷)

Theoremsyl5eqbr 3788 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
A = B    &   (φB𝑅𝐶)       (φA𝑅𝐶)

Theoremsyl5eqbrr 3789 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
B = A    &   (φB𝑅𝐶)       (φA𝑅𝐶)

Theoremsyl5breq 3790 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
A𝑅B    &   (φB = 𝐶)       (φA𝑅𝐶)

Theoremsyl5breqr 3791 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
A𝑅B    &   (φ𝐶 = B)       (φA𝑅𝐶)

Theoremsyl6eqbr 3792 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
(φA = B)    &   B𝑅𝐶       (φA𝑅𝐶)

Theoremsyl6eqbrr 3793 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
(φB = A)    &   B𝑅𝐶       (φA𝑅𝐶)

Theoremsyl6breq 3794 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
(φA𝑅B)    &   B = 𝐶       (φA𝑅𝐶)

Theoremsyl6breqr 3795 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
(φA𝑅B)    &   𝐶 = B       (φA𝑅𝐶)

Theoremssbrd 3796 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
(φAB)       (φ → (𝐶A𝐷𝐶B𝐷))

Theoremssbri 3797 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
AB       (𝐶A𝐷𝐶B𝐷)

Theoremnfbrd 3798 Deduction version of bound-variable hypothesis builder nfbr 3799. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
(φxA)    &   (φx𝑅)    &   (φxB)       (φ → Ⅎx A𝑅B)

Theoremnfbr 3799 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
xA    &   x𝑅    &   xB       x A𝑅B

Theorembrab1 3800* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
(x𝑅Ax {zz𝑅A})

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