P. 40 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eluni 3901*	Membership in class union. (Contributed by NM, 22-May-1994.)
|- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
Theorem	eluni2 3902*	Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
|- ( A e. U. B <-> E. x e. B A e. x )
Theorem	elunii 3903	Membership in class union. (Contributed by NM, 24-Mar-1995.)
|- ( ( A e. B /\ B e. C ) -> A e. U. C )
Theorem	nfuni 3904	Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- F/_ x A   =>    |- F/_ x U. A
Theorem	nfunid 3905	Deduction version of nfuni 3904. (Contributed by NM, 18-Feb-2013.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x U. A )
Theorem	csbunig 3906	Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> [_ A / x ]_ U. B = U. [_ A / x ]_ B )
Theorem	unieq 3907	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 3908	Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
|- A = B   =>    |- U. A = U. B
Theorem	unieqd 3909	Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> U. A = U. B )
Theorem	eluniab 3910*	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 3911*	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 3912	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 3913	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 3914	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 3915	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 3916*	An alternate 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/_ x A <-> A. y F/ x y = A ) )
Theorem	uniun 3917	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 3918	The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (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 3919	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 3920	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 3921	Subclass relationship for subclass union. Inference form of uniss 3919. (Contributed by David Moews, 1-May-2017.)
|- A C_ B   =>    |- U. A C_ U. B
Theorem	unissd 3922	Subclass relationship for subclass union. Deduction form of uniss 3919. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   =>    |- ( ph -> U. A C_ U. B )
Theorem	uni0b 3923	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 3924*	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 3925	The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
|- U. (/) = (/)
Theorem	elssuni 3926	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 3927	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 3928*	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 3929*	A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)
|- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B )
Theorem	unidif 3930*	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 3931*	Relationship implying union. (Contributed by NM, 10-Nov-1999.)
|- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B )
Theorem	unimax 3932*	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 )
2.1.19  The intersection of a class
Syntax	cint 3933	Extend class notation to include the intersection of a class. Read: "intersection (of) A".
class |^| A
Definition	df-int 3934*	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 3207. (Contributed by NM, 18-Aug-1993.)
|- |^| A = { x | A. y ( y e. A -> x e. y ) }
Theorem	dfint2 3935*	Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
|- |^| A = { x | A. y e. A x e. y }
Theorem	inteq 3936	Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
|- ( A = B -> |^| A = |^| B )
Theorem	inteqi 3937	Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
|- A = B   =>    |- |^| A = |^| B
Theorem	inteqd 3938	Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
|- ( ph -> A = B )   =>    |- ( ph -> |^| A = |^| B )
Theorem	elint 3939*	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 3940*	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 3941*	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 3942	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 3943	Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- F/_ x A   =>    |- F/_ x |^| A
Theorem	elintab 3944*	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 3945*	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 3946*	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 3947	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 3948	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 3949*	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 3950*	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 3951*	Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|- A C_ |^| { x e. B | A C_ x }
Theorem	ssmin 3952*	Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
|- A C_ |^| { x | ( A C_ x /\ ph ) }
Theorem	intmin 3953*	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 3954	Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B -> |^| B C_ |^| A )
Theorem	intssunim 3955*	The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
|- ( E. x x e. A -> |^| A C_ U. A )
Theorem	ssintrab 3956*	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	intssuni2m 3957*	Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B )
Theorem	intminss 3958*	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 3959*	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 3960*	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 3961*	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 3962*	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 ). (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 3963	The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
|- ( (/) e. A -> |^| A = (/) )
Theorem	intun 3964	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 3965	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 3966	The intersection of a pair is the intersection of its members. Closed form of intpr 3965. 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 3967	Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A e. V -> |^| { A } = A )
Theorem	intsn 3968	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	uniintsnr 3969*	The union and intersection of a singleton are equal. See also eusn 3749. (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( E. x A = { x } -> U. A = |^| A )
Theorem	uniintabim 3970	The union and the intersection of a class abstraction are equal if there is a unique satisfying value of ph ( x ). (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( E! x ph -> U. { x | ph } = |^| { x | ph } )
Theorem	intunsn 3971	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 3972	Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X = (/) -> ( A i^i |^| X ) = A )
Theorem	elrint 3973*	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 3974*	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 ) )
2.1.20  Indexed union and intersection
Syntax	ciun 3975	Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation U. x e. A B, with the same union symbol as cuni 3898. 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 3976	Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation |^| x e. A B, with the same intersection symbol as cint 3933. 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 3977*	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 disjoint variable group (meaning A cannot depend on x) and that B and x do not share a disjoint 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 4009. Theorem uniiun 4029 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)
|- U_ x e. A B = { y | E. x e. A y e. B }
Definition	df-iin 3978*	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3977. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4010. Theorem intiin 4030 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 3979*	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 3980*	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 3981*	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 3982	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	iunconstm 3983*	Indexed union of a constant class, i.e. where B does not depend on x. (Contributed by Jim Kingdon, 15-Aug-2018.)
|- ( E. x x e. A -> U_ x e. A B = B )
Theorem	iinconstm 3984*	Indexed intersection of a constant class, i.e. where B does not depend on x. (Contributed by Jim Kingdon, 19-Dec-2018.)
|- ( E. y y e. A -> |^|_ x e. A B = B )
Theorem	iuniin 3985*	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 3986*	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 3987*	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 3988*	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 3989*	Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
|- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )
Theorem	ss2iun 3990	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 3991	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 3992	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 3993	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 3994	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 3995	Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> |^|_ x e. A B = |^|_ x e. A C )
Theorem	iuneq2dv 3996*	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 3997*	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 3998*	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 3999*	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 4000*	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 )