P. 40 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 3901-4000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	unieq 3901	Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- (A = B -> U.A = U.B)
Theorem	unieqi 3902	Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
|- A = B   =>   |- U.A = U.B
Theorem	unieqd 3903	Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
|- (ph -> A = B)   =>   |- (ph -> U.A = U.B)
Theorem	eluniab 3904*	Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
|- (A e. U.{x | ph} <-> E.x(A e. x /\ ph))
Theorem	elunirab 3905*	Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
|- (A e. U.{x e. B | ph} <-> E.x e. B (A e. x /\ ph))
Theorem	unipr 3906	The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
|- A e. _V   &   |- B e. _V   =>   |- U.{A, B} = (A u. B)
Theorem	uniprg 3907	The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
|- ((A e. V /\ B e. W) -> U.{A, B} = (A u. B))
Theorem	unisn 3908	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>   |- U.{A} = A
Theorem	unisng 3909	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
|- (A e. V -> U.{A} = A)
Theorem	dfnfc2 3910*	An alternative statement of the effective freeness of a class A, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- (A.x A e. V -> ( F/_xA <-> A.y F/x y = A))
Theorem	uniun 3911	The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)
|- U.(A u. B) = (U.A u. U.B)
Theorem	uniin 3912	The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs in set.mm for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- U.(A i^i B) C_ (U.A i^i U.B)
Theorem	uniss 3913	Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- (A C_ B -> U.A C_ U.B)
Theorem	ssuni 3914	Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ((A C_ B /\ B e. C) -> A C_ U.C)
Theorem	unissi 3915	Subclass relationship for subclass union. Inference form of uniss 3913. (Contributed by David Moews, 1-May-2017.)
|- A C_ B   =>   |- U.A C_ U.B
Theorem	unissd 3916	Subclass relationship for subclass union. Deduction form of uniss 3913. (Contributed by David Moews, 1-May-2017.)
|- (ph -> A C_ B)   =>   |- (ph -> U.A C_ U.B)
Theorem	uni0b 3917	The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)
|- (U.A = (/) <-> A C_ {(/)})
Theorem	uni0c 3918*	The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
|- (U.A = (/) <-> A.x e. A x = (/))
Theorem	uni0 3919	The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul in set.mm by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
|- U.(/) = (/)
Theorem	elssuni 3920	An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)
|- (A e. B -> A C_ U.B)
Theorem	unissel 3921	Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
|- ((U.A C_ B /\ B e. A) -> U.A = B)
Theorem	unissb 3922*	Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
|- (U.A C_ B <-> A.x e. A x C_ B)
Theorem	uniss2 3923*	A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 4012 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)
|- (A.x e. A E.y e. B x C_ y -> U.A C_ U.B)
Theorem	unidif 3924*	If the difference A \ B contains the largest members of A, then the union of the difference is the union of A. (Contributed by NM, 22-Mar-2004.)
|- (A.x e. A E.y e. (A \ B)x C_ y -> U.(A \ B) = U.A)
Theorem	ssunieq 3925*	Relationship implying union. (Contributed by NM, 10-Nov-1999.)
|- ((A e. B /\ A.x e. B x C_ A) -> A = U.B)
Theorem	unimax 3926*	Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
|- (A e. B -> U.{x e. B | x C_ A} = A)
2.1.18  The intersection of a class
Syntax	cint 3927	Extend class notation to include the intersection of a class (read: 'intersect A').
class |^|A
Definition	df-int 3928*	Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example, |^|{{ 1 , 3 }, { 1 , 8 }} = { 1 }. Compare this with the intersection of two classes, df-in 3214. (Contributed by NM, 18-Aug-1993.)
|- |^|A = {x | A.y(y e. A -> x e. y)}
Theorem	dfint2 3929*	Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
|- |^|A = {x | A.y e. A x e. y}
Theorem	inteq 3930	Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
|- (A = B -> |^|A = |^|B)
Theorem	inteqi 3931	Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
|- A = B   =>   |- |^|A = |^|B
Theorem	inteqd 3932	Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
|- (ph -> A = B)   =>   |- (ph -> |^|A = |^|B)
Theorem	elint 3933*	Membership in class intersection. (Contributed by NM, 21-May-1994.)
|- A e. _V   =>   |- (A e. |^|B <-> A.x(x e. B -> A e. x))
Theorem	elint2 3934*	Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
|- A e. _V   =>   |- (A e. |^|B <-> A.x e. B A e. x)
Theorem	elintg 3935*	Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
|- (A e. V -> (A e. |^|B <-> A.x e. B A e. x))
Theorem	elinti 3936	Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- (A e. |^|B -> (C e. B -> A e. C))
Theorem	nfint 3937	Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- F/_xA   =>   |- F/_x|^|A
Theorem	elintab 3938*	Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>   |- (A e. |^|{x | ph} <-> A.x(ph -> A e. x))
Theorem	elintrab 3939*	Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
|- A e. _V   =>   |- (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x))
Theorem	elintrabg 3940*	Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
|- (A e. V -> (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x)))
Theorem	int0 3941	The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
|- |^|(/) = _V
Theorem	intss1 3942	An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)
|- (A e. B -> |^|B C_ A)
Theorem	ssint 3943*	Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
|- (A C_ |^|B <-> A.x e. B A C_ x)
Theorem	ssintab 3944*	Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- (A C_ |^|{x | ph} <-> A.x(ph -> A C_ x))
Theorem	ssintub 3945*	Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|- A C_ |^|{x e. B | A C_ x}
Theorem	ssmin 3946*	Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
|- A C_ |^|{x | (A C_ x /\ ph)}
Theorem	intmin 3947*	Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- (A e. B -> |^|{x e. B | A C_ x} = A)
Theorem	intss 3948	Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|- (A C_ B -> |^|B C_ |^|A)
Theorem	intssuni 3949	The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
|- (A =/= (/) -> |^|A C_ U.A)
Theorem	ssintrab 3950*	Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
|- (A C_ |^|{x e. B | ph} <-> A.x e. B (ph -> A C_ x))
Theorem	unissint 3951	If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 3964). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- (U.A C_ |^|A <-> (A = (/) \/ U.A = |^|A))
Theorem	intssuni2 3952	Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
|- ((A C_ B /\ A =/= (/)) -> |^|A C_ U.B)
Theorem	intminss 3953*	Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)
|- (x = A -> (ph <-> ps))   =>   |- ((A e. B /\ ps) -> |^|{x e. B | ph} C_ A)
Theorem	intmin2 3954*	Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
|- A e. _V   =>   |- |^|{x | A C_ x} = A
Theorem	intmin3 3955*	Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)
|- (x = A -> (ph <-> ps))   &   |- ps   =>   |- (A e. V -> |^|{x | ph} C_ A)
Theorem	intmin4 3956*	Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
|- (A C_ |^|{x | ph} -> |^|{x | (A C_ x /\ ph)} = |^|{x | ph})
Theorem	intab 3957*	The intersection of a special case of a class abstraction. y may be free in ph and A, which can be thought of a ph(y) and A(y). Typically, abrexex2 in set.mm or abexssex in set.mm can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
|- A e. _V   &   |- {x | E.y(ph /\ x = A)} e. _V   =>   |- |^|{x | A.y(ph -> A e. x)} = {x | E.y(ph /\ x = A)}
Theorem	int0el 3958	The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
|- ((/) e. A -> |^|A = (/))
Theorem	intun 3959	The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
|- |^|(A u. B) = (|^|A i^i |^|B)
Theorem	intpr 3960	The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
|- A e. _V   &   |- B e. _V   =>   |- |^|{A, B} = (A i^i B)
Theorem	intprg 3961	The intersection of a pair is the intersection of its members. Closed form of intpr 3960. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
|- ((A e. V /\ B e. W) -> |^|{A, B} = (A i^i B))
Theorem	intsng 3962	Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- (A e. V -> |^|{A} = A)
Theorem	intsn 3963	The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
|- A e. _V   =>   |- |^|{A} = A
Theorem	uniintsn 3964*	Two ways to express "A is a singleton." See also en1 in set.mm, en1b in set.mm, card1 in set.mm, and eusn 3797. (Contributed by NM, 2-Aug-2010.)
|- (U.A = |^|A <-> E.x A = {x})
Theorem	uniintab 3965	The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of ph(x). (Contributed by Mario Carneiro, 24-Dec-2016.)
|- (E!xph <-> U.{x | ph} = |^|{x | ph})
Theorem	intunsn 3966	Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
|- B e. _V   =>   |- |^|(A u. {B}) = (|^|A i^i B)
Theorem	rint0 3967	Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- (X = (/) -> (A i^i |^|X) = A)
Theorem	elrint 3968*	Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- (X e. (A i^i |^|B) <-> (X e. A /\ A.y e. B X e. y))
Theorem	elrint2 3969*	Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- (X e. A -> (X e. (A i^i |^|B) <-> A.y e. B X e. y))
2.1.19  Indexed union and intersection
Syntax	ciun 3970	Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation U.x e. AB, with the same union symbol as cuni 3892. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses as distinguished symbol U_ instead of U. and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class U_x e. A B
Syntax	ciin 3971	Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation |^|x e. AB, with the same intersection symbol as cint 3927. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol |^|_ instead of |^| and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class |^|_x e. A B
Definition	df-iun 3972*	Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, A is independent of x (although this is not required by the definition), and B depends on x i.e. can be read informally as B(x). We call x the index, A the index set, and B the indexed set. In most books, x e. A is written as a subscript or underneath a union symbol U.. We use a special union symbol U_ to make it easier to distinguish from plain class union. In many theorems, you will see that x and A are in the same distinct variable group (meaning A cannot depend on x) and that B and x do not share a distinct variable group (meaning that can be thought of as B(x) i.e. can be substituted with a class expression containing x). An alternate definition tying indexed union to ordinary union is dfiun2 4002. Theorem uniiun 4020 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5467 and funiunfv 5468 are useful when B is a function. (Contributed by NM, 27-Jun-1998.)
|- U_x e. A B = {y | E.x e. A y e. B}
Definition	df-iin 3973*	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3972. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4003. Theorem intiin 4021 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
|- |^|_x e. A B = {y | A.x e. A y e. B}
Theorem	eliun 3974*	Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
|- (A e. U_x e. B C <-> E.x e. B A e. C)
Theorem	eliin 3975*	Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
|- (A e. V -> (A e. |^|_x e. B C <-> A.x e. B A e. C))
Theorem	iuncom 3976*	Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
|- U_x e. A U_y e. B C = U_y e. B U_x e. A C
Theorem	iuncom4 3977	Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
|- U_x e. A U.B = U.U_x e. A B
Theorem	iunconst 3978*	Indexed union of a constant class, i.e. where B does not depend on x. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- (A =/= (/) -> U_x e. A B = B)
Theorem	iinconst 3979*	Indexed intersection of a constant class, i.e. where B does not depend on x. (Contributed by Mario Carneiro, 6-Feb-2015.)
|- (A =/= (/) -> |^|_x e. A B = B)
Theorem	iuniin 3980*	Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- U_x e. A |^|_y e. B C C_ |^|_y e. B U_x e. A C
Theorem	iunss1 3981*	Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- (A C_ B -> U_x e. A C C_ U_x e. B C)
Theorem	iinss1 3982*	Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
|- (A C_ B -> |^|_x e. B C C_ |^|_x e. A C)
Theorem	iuneq1 3983*	Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
|- (A = B -> U_x e. A C = U_x e. B C)
Theorem	iineq1 3984*	Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
|- (A = B -> |^|_x e. A C = |^|_x e. B C)
Theorem	ss2iun 3985	Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- (A.x e. A B C_ C -> U_x e. A B C_ U_x e. A C)
Theorem	iuneq2 3986	Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
|- (A.x e. A B = C -> U_x e. A B = U_x e. A C)
Theorem	iineq2 3987	Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- (A.x e. A B = C -> |^|_x e. A B = |^|_x e. A C)
Theorem	iuneq2i 3988	Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
|- (x e. A -> B = C)   =>   |- U_x e. A B = U_x e. A C
Theorem	iineq2i 3989	Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
|- (x e. A -> B = C)   =>   |- |^|_x e. A B = |^|_x e. A C
Theorem	iineq2d 3990	Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
|- F/xph   &   |- ((ph /\ x e. A) -> B = C)   =>   |- (ph -> |^|_x e. A B = |^|_x e. A C)
Theorem	iuneq2dv 3991*	Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
|- ((ph /\ x e. A) -> B = C)   =>   |- (ph -> U_x e. A B = U_x e. A C)
Theorem	iineq2dv 3992*	Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
|- ((ph /\ x e. A) -> B = C)   =>   |- (ph -> |^|_x e. A B = |^|_x e. A C)
Theorem	iuneq1d 3993*	Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
|- (ph -> A = B)   =>   |- (ph -> U_x e. A C = U_x e. B C)
Theorem	iuneq12d 3994*	Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> U_x e. A C = U_x e. B D)
Theorem	iuneq2d 3995*	Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
|- (ph -> B = C)   =>   |- (ph -> U_x e. A B = U_x e. A C)
Theorem	nfiun 3996	Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_yA   &   |- F/_yB   =>   |- F/_yU_x e. A B
Theorem	nfiin 3997	Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_yA   &   |- F/_yB   =>   |- F/_y|^|_x e. A B
Theorem	nfiu1 3998	Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
|- F/_xU_x e. A B
Theorem	nfii1 3999	Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
|- F/_x|^|_x e. A B
Theorem	dfiun2g 4000*	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.)
|- (A.x e. A B e. C -> U_x e. A B = U.{y | E.x e. A y = B})