mmtheorems26 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2501-2600 - Page 26 of 107

Type	Label	Description
Statement
The union of a class
Syntax	cuni 2501	Extend class notation to include the union of a class (read: 'union A ')
class U.A
Definition	df-uni 2502	Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of [TakeutiZaring] p. 16.
|- U.A = {x | E.y(x e. y /\ y e. A)}
Theorem	dfuni2 2503	Alternate definition of class union.
|- U.A = {x | E.y e. A x e. y}
Theorem	eluni 2504	Membership in class union.
|- (A e. U.B <-> E.x(A e. x /\ x e. B))
Theorem	eluni2 2505	Membership in class union. Restricted quantifier version.
|- (A e. U.B <-> E.x e. B A e. x)
Theorem	elunii 2506	Membership in class union.
|- ((A e. B /\ B e. C) -> A e. U.C)
Theorem	hbuni 2507	Bound-variable hypothesis builder for union.
|- (y e. A -> A.x y e. A)   =>   |- (y e. U.A -> A.x y e. U.A)
Theorem	unieq 2508	Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.
|- (A = B -> U.A = U.B)
Theorem	unieqi 2509	Inference of equality of two class unions.
|- A = B   =>   |- U.A = U.B
Theorem	unieqd 2510	Deduction of equality of two class unions.
|- (ph -> A = B)   =>   |- (ph -> U.A = U.B)
Theorem	eluniab 2511	Membership in union of a class abstraction.
|- (A e. U.{x | ph} <-> E.x(A e. x /\ ph))
Theorem	elunirab 2512	Membership in union of a class abstraction.
|- (A e. U.{x e. B | ph} <-> E.x e. B (A e. x /\ ph))
Theorem	unipr 2513	The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16.
|- A e. V   &   |- B e. V   =>   |- U.{A, B} = (A u. B)
Theorem	uniprg 2514	The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16.
|- ((A e. C /\ B e. D) -> U.{A, B} = (A u. B))
Theorem	unisn 2515	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|- A e. V   =>   |- U.{A} = A
Theorem	unisng 2516	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|- (A e. B -> U.{A} = A)
Theorem	uniun 2517	The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53.
|- U.(A u. B) = (U.A u. U.B)
Theorem	uniin 2518	The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235.
|- U.(A i^i B) (_ (U.A i^i U.B)
Theorem	uniss 2519	Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.
|- (A (_ B -> U.A (_ U.B)
Theorem	ssuni 2520	Subclass relationship for class union.
|- ((A (_ B /\ B e. C) -> A (_ U.C)
Theorem	uni0b 2521	The union of a set is empty iff the set is included in the singleton of the empty set.
|- (U.A = (/) <-> A (_ {(/)})
Theorem	uni0c 2522	The union of a set is empty iff all of its members are empty.
|- (U.A = (/) <-> A.x e. A x = (/))
Theorem	uni0 2523	The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 2708 by Eric Schmidt, 4-Apr-2007.)
|- U.(/) = (/)
Theorem	elssuni 2524	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.
|- (A e. B -> A (_ U.B)
Theorem	unissel 2525	Condition turning a subclass relationship for union into an equality.
|- ((U.A (_ B /\ B e. A) -> U.A = B)
Theorem	unissb 2526	Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse.
|- (U.A (_ B <-> A.x e. A x (_ B)
Theorem	uniss2 2527	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 2593 for a generalization to indexed unions.
|- (A.x e. A E.y e. B x (_ y -> U.A (_ U.B)
Theorem	unidif 2528	If the difference A \ B contains the largest members of A, then the union of the difference is the union of A.
|- (A.x e. A E.y e. (A \ B)x (_ y -> U.(A \ B) = U.A)
Theorem	ssunieq 2529	Relationship implying union.
|- ((A e. B /\ A.x e. B x (_ A) -> A = U.B)
Theorem	unimax 2530	Any member of a class is the largest of those members that it includes.
|- (A e. B -> U.{x e. B | x (_ A} = A)
The intersection of a class
Syntax	cint 2531	Extend class notation to include the intersection of a class (read: 'intersect A ').
class |^|A
Definition	df-int 2532	Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44.
|- |^|A = {x | A.y(y e. A -> x e. y)}
Theorem	dfint2 2533	Alternate definition of class intersection.
|- |^|A = {x | A.y e. A x e. y}
Theorem	inteq 2534	Equality law for intersection.
|- (A = B -> |^|A = |^|B)
Theorem	inteqi 2535	Equality inference for class intersection.
|- A = B   =>   |- |^|A = |^|B
Theorem	inteqd 2536	Equality deduction for class intersection.
|- (ph -> A = B)   =>   |- (ph -> |^|A = |^|B)
Theorem	elint 2537	Membership in class intersection.
|- A e. V   =>   |- (A e. |^|B <-> A.x(x e. B -> A e. x))
Theorem	elint2 2538	Membership in class intersection.
|- A e. V   =>   |- (A e. |^|B <-> A.x e. B A e. x)
Theorem	elintg 2539	Membership in class intersection, with the sethood requirement expressed as an antecedent.
|- (A e. C -> (A e. |^|B <-> A.x e. B A e. x))
Theorem	elinti 2540	Membership in class intersection.
|- (A e. |^|B -> (C e. B -> A e. C))
Theorem	hbint 2541	Bound-variable hypothesis builder for intersection.
|- (y e. A -> A.x y e. A)   =>   |- (y e. |^|A -> A.x y e. |^|A)
Theorem	elintab 2542	Membership in the intersection of a class abstraction.
|- A e. V   =>   |- (A e. |^|{x | ph} <-> A.x(ph -> A e. x))
Theorem	elintrab 2543	Membership in the intersection of a class abstraction.
|- A e. V   =>   |- (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x))
Theorem	elintrabg 2544	Membership in the intersection of a class abstraction.
|- (A e. C -> (A e. |^|{x e. B | ph} <-> A.x e. B (ph -> A e. x)))
Theorem	int0 2545	The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44.
|- |^|(/) = V
Theorem	intss1 2546	An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes.
|- (A e. B -> |^|B (_ A)
Theorem	ssint 2547	Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse.
|- (A (_ |^|B <-> A.x e. B A (_ x)
Theorem	ssintab 2548	Subclass of the intersection of a class abstraction.
|- (A (_ |^|{x | ph} <-> A.x(ph -> A (_ x))
Theorem	ssintub 2549	Subclass of a least upper bound.
|- A (_ |^|{x e. B | A (_ x}
Theorem	ssmin 2550	Subclass of the minimum value of class of supersets.
|- A (_ |^|{x | (A (_ x /\ ph)}
Theorem	intmin 2551	Any member of a class is the smallest of those members that include it.
|- (A e. B -> |^|{x e. B | A (_ x} = A)
Theorem	intss 2552	Intersection of subclasses.
|- (A (_ B -> |^|B (_ |^|A)
Theorem	intssuni 2553	The intersection of a nonempty set is a subclass of its union.
|- (A =/= (/) -> |^|A (_ U.A)
Theorem	intssuni2 2554	Subclass relationship for intersection and union.
|- ((A (_ B /\ A =/= (/)) -> |^|A (_ U.B)
Theorem	intmin2 2555	Any set is the smallest of all sets that include it.
|- A e. V   =>   |- |^|{x | A (_ x} = A
Theorem	intmin3 2556	Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members.
|- (x = A -> (ph <-> ps))   &   |- ps   =>   |- (A e. B -> |^|{x | ph} (_ A)
Theorem	intmin4 2557	Elimination of a conjunct in a class intersection.
|- (A (_ |^|{x | ph} -> |^|{x | (A (_ x /\ ph)} = |^|{x | ph})
Theorem	intab 2558	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 3869 or abexssex 3870 can be used to satisfy the second hypothesis.
|- 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 2559	The intersection of a class containing the empty set is empty.
|- ((/) e. A -> |^|A = (/))
Theorem	intun 2560	The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42.
|- |^|(A u. B) = (|^|A i^i |^|B)
Theorem	intpr 2561	The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42.
|- A e. V   &   |- B e. V   =>   |- |^|{A, B} = (A i^i B)
Theorem	intsn 2562	The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41.
|- A e. V   =>   |- |^|{A} = A
Theorem	intunsn 2563	Theorem joining a singleton to an intersection.
|- B e. V   =>   |- |^|(A u. {B}) = (|^|A i^i B)
Indexed union and intersection
Syntax	ciun 2564	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 2501. 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 2565	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 2531. 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 2566	Define indexed union. Definition of [Stoll] p. 45. In normal use, A is independent of x, 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 2585. Theorem uniiun 2599 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 3863 and funiunfv 3864 are useful when B is a function.
|- U_x e. A B = {y | E.x e. A y e. B}
Definition	df-iin 2567	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 2566. An alternate definition tying indexed intersection to ordinary intersection is dfiun2 2585. Theorem intiin 2600 provides a definition of ordinary intersection in terms of indexed intersection.
|- |^|_x e. A B = {y | A.x e. A y e. B}
Theorem	eliun 2568	Membership in indexed union.
|- (A e. U_x e. B C <-> E.x e. B A e. C)
Theorem	eliin 2569	Membership in indexed intersection.
|- (A e. D -> (A e. |^|_x e. B C <-> A.x e. B A e. C))
Theorem	iunconst 2570	Indexed union of a constant class, i.e. where B does not depend on x.
|- (A =/= (/) -> U_x e. A B = B)
Theorem	iuniin 2571	Law combining indexed union with indexed intersection. (This theorem appears as the last example on http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. If anyone has a literature reference, please inform N. Megill.)
|- U_x e. A |^|_y e. B C (_ |^|_y e. B U_x e. A C
Theorem	iunss1 2572	Subclass theorem for indexed union.
|- (A (_ B -> U_x e. A C (_ U_x e. B C)
Theorem	iuneq1 2573	Equality theorem for indexed union.
|- (A = B -> U_x e. A C = U_x e. B C)
Theorem	iineq1 2574	Equality theorem for restricted existential quantifier.
|- (A = B -> |^|_x e. A C = |^|_x e. B C)
Theorem	ss2iun 2575	Subclass theorem for indexed union.
|- (A.x e. A B (_ C ->