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Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiunxun 3701 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 3702* 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 3703* 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 3704* 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 3705 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(A ⊆ 𝒫 B AB)
 
Theorempwssb 3706* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
(A ⊆ 𝒫 Bx A xB)
 
Theoremelpwuni 3707 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(B A → (A ⊆ 𝒫 B A = B))
 
Theoremiinpw 3708* 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 3709* 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 3710* 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 3711 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 3712* 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 3713* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
(Disj x A By∃*x(x A y B))
 
Theoremdisjss2 3714 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 3715 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(x A B = 𝐶 → (Disj x A BDisj x A 𝐶))
 
Theoremdisjeq2dv 3716* 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 3717* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(AB → (Disj x B 𝐶Disj x A 𝐶))
 
Theoremdisjeq1 3718* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(A = B → (Disj x A 𝐶Disj x B 𝐶))
 
Theoremdisjeq1d 3719* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(φA = B)       (φ → (Disj x A 𝐶Disj x B 𝐶))
 
Theoremdisjeq12d 3720* 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 3721* 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 3722* 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 3723* Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
yA    &   yB       yDisj x A B
 
Theoremnfdisj1 3724 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
xDisj x A B
 
Theoreminvdisj 3725* 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 3726 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x A {x}
 
Theorem0disj 3727 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x A
 
Theoremdisjxsn 3728* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x {A}B
 
Theoremdisjx0 3729 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj x B
 
2.1.22  Binary relations
 
Syntaxwbr 3730 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 3731 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 4518). (Contributed by NM, 31-Dec-1993.)
(A𝑅B ↔ ⟨A, B 𝑅)
 
Theorembreq 3732 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
(𝑅 = 𝑆 → (A𝑅BA𝑆B))
 
Theorembreq1 3733 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(A = B → (A𝑅𝐶B𝑅𝐶))
 
Theorembreq2 3734 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(A = B → (𝐶𝑅A𝐶𝑅B))
 
Theorembreq12 3735 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
((A = B 𝐶 = 𝐷) → (A𝑅𝐶B𝑅𝐷))
 
Theorembreqi 3736 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
𝑅 = 𝑆       (A𝑅BA𝑆B)
 
Theorembreq1i 3737 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
A = B       (A𝑅𝐶B𝑅𝐶)
 
Theorembreq2i 3738 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
A = B       (𝐶𝑅A𝐶𝑅B)
 
Theorembreq12i 3739 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 3740 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)       (φ → (A𝑅𝐶B𝑅𝐶))
 
Theorembreqd 3741 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(φA = B)       (φ → (𝐶A𝐷𝐶B𝐷))
 
Theorembreq2d 3742 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)       (φ → (𝐶𝑅A𝐶𝑅B))
 
Theorembreq12d 3743 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 3744 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(φA = B)    &   (φ𝑅 = 𝑆)    &   (φ𝐶 = 𝐷)       (φ → (A𝑅𝐶B𝑆𝐷))
 
Theorembreqan12d 3745 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)    &   (ψ𝐶 = 𝐷)       ((φ ψ) → (A𝑅𝐶B𝑅𝐷))
 
Theorembreqan12rd 3746 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(φA = B)    &   (ψ𝐶 = 𝐷)       ((ψ φ) → (A𝑅𝐶B𝑅𝐷))
 
Theoremnbrne1 3747 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 3748 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 3749 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   B𝑅𝐶       A𝑅𝐶
 
Theoremeqbrtrd 3750 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
(φA = B)    &   (φB𝑅𝐶)       (φA𝑅𝐶)
 
Theoremeqbrtrri 3751 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   A𝑅𝐶       B𝑅𝐶
 
Theoremeqbrtrrd 3752 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(φA = B)    &   (φA𝑅𝐶)       (φB𝑅𝐶)
 
Theorembreqtri 3753 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A𝑅B    &   B = 𝐶       A𝑅𝐶
 
Theorembreqtrd 3754 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(φA𝑅B)    &   (φB = 𝐶)       (φA𝑅𝐶)
 
Theorembreqtrri 3755 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
A𝑅B    &   𝐶 = B       A𝑅𝐶
 
Theorembreqtrrd 3756 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(φA𝑅B)    &   (φ𝐶 = B)       (φA𝑅𝐶)
 
Theorem3brtr3i 3757 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
A𝑅B    &   A = 𝐶    &   B = 𝐷       𝐶𝑅𝐷
 
Theorem3brtr4i 3758 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
A𝑅B    &   𝐶 = A    &   𝐷 = B       𝐶𝑅𝐷
 
Theorem3brtr3d 3759 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
(φA𝑅B)    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐶𝑅𝐷)
 
Theorem3brtr4d 3760 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
(φA𝑅B)    &   (φ𝐶 = A)    &   (φ𝐷 = B)       (φ𝐶𝑅𝐷)
 
Theorem3brtr3g 3761 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(φA𝑅B)    &   A = 𝐶    &   B = 𝐷       (φ𝐶𝑅𝐷)
 
Theorem3brtr4g 3762 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(φA𝑅B)    &   𝐶 = A    &   𝐷 = B       (φ𝐶𝑅𝐷)
 
Theoremsyl5eqbr 3763 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
A = B    &   (φB𝑅𝐶)       (φA𝑅𝐶)
 
Theoremsyl5eqbrr 3764 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
B = A    &   (φB𝑅𝐶)       (φA𝑅𝐶)
 
Theoremsyl5breq 3765 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
A𝑅B    &   (φB = 𝐶)       (φA𝑅𝐶)
 
Theoremsyl5breqr 3766 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
A𝑅B    &   (φ𝐶 = B)       (φA𝑅𝐶)
 
Theoremsyl6eqbr 3767 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
(φA = B)    &   B𝑅𝐶       (φA𝑅𝐶)
 
Theoremsyl6eqbrr 3768 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
(φB = A)    &   B𝑅𝐶       (φA𝑅𝐶)
 
Theoremsyl6breq 3769 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
(φA𝑅B)    &   B = 𝐶       (φA𝑅𝐶)
 
Theoremsyl6breqr 3770 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
(φA𝑅B)    &   𝐶 = B       (φA𝑅𝐶)
 
Theoremssbrd 3771 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
(φAB)       (φ → (𝐶A𝐷𝐶B𝐷))
 
Theoremssbri 3772 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
AB       (𝐶A𝐷𝐶B𝐷)
 
Theoremnfbrd 3773 Deduction version of bound-variable hypothesis builder nfbr 3774. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
(φxA)    &   (φx𝑅)    &   (φxB)       (φ → Ⅎx A𝑅B)
 
Theoremnfbr 3774 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 3775* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
(x𝑅Ax {zz𝑅A})
 
Theorembrun 3776 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
(A(𝑅𝑆)B ↔ (A𝑅B A𝑆B))
 
Theorembrin 3777 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
(A(𝑅𝑆)B ↔ (A𝑅B A𝑆B))
 
Theorembrdif 3778 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
(A(𝑅𝑆)B ↔ (A𝑅B ¬ A𝑆B))
 
Theoremsbcbrg 3779 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(A 𝐷 → ([A / x]B𝑅𝐶A / xBA / x𝑅A / x𝐶))
 
Theoremsbcbr12g 3780* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(A 𝐷 → ([A / x]B𝑅𝐶A / xB𝑅A / x𝐶))
 
Theoremsbcbr1g 3781* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(A 𝐷 → ([A / x]B𝑅𝐶A / xB𝑅𝐶))
 
Theoremsbcbr2g 3782* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(A 𝐷 → ([A / x]B𝑅𝐶B𝑅A / x𝐶))
 
2.1.23  Ordered-pair class abstractions (class builders)
 
Syntaxcopab 3783 Extend class notation to include ordered-pair class abstraction (class builder).
class {⟨x, y⟩ ∣ φ}
 
Syntaxcmpt 3784 Extend the definition of a class to include maps-to notation for defining a function via a rule.
class (x AB)
 
Definitiondf-opab 3785* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually x and y are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)
{⟨x, y⟩ ∣ φ} = {zxy(z = ⟨x, y φ)}
 
Definitiondf-mpt 3786* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from x (in A) to B(x)." The class expression B is the value of the function at x and normally contains the variable x. Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)
(x AB) = {⟨x, y⟩ ∣ (x A y = B)}
 
Theoremopabss 3787* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{⟨x, y⟩ ∣ x𝑅y} ⊆ 𝑅
 
Theoremopabbid 3788 Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
xφ    &   yφ    &   (φ → (ψχ))       (φ → {⟨x, y⟩ ∣ ψ} = {⟨x, y⟩ ∣ χ})
 
Theoremopabbidv 3789* Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)
(φ → (ψχ))       (φ → {⟨x, y⟩ ∣ ψ} = {⟨x, y⟩ ∣ χ})
 
Theoremopabbii 3790 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
(φψ)       {⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ}
 
Theoremnfopab 3791* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)
zφ       z{⟨x, y⟩ ∣ φ}
 
Theoremnfopab1 3792 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
x{⟨x, y⟩ ∣ φ}
 
Theoremnfopab2 3793 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
y{⟨x, y⟩ ∣ φ}
 
Theoremcbvopab 3794* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
zφ    &   wφ    &   xψ    &   yψ    &   ((x = z y = w) → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨z, w⟩ ∣ ψ}
 
Theoremcbvopabv 3795* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
((x = z y = w) → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨z, w⟩ ∣ ψ}
 
Theoremcbvopab1 3796* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
zφ    &   xψ    &   (x = z → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}
 
Theoremcbvopab2 3797* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
zφ    &   yψ    &   (y = z → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}
 
Theoremcbvopab1s 3798* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
{⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ [z / x]φ}
 
Theoremcbvopab1v 3799* Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
(x = z → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}
 
Theoremcbvopab2v 3800* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
(y = z → (φψ))       {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}
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