mmtheorems26 - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	dfuni2 2501	Alternate definition of class union.
|- U.A = {x | E.y e. A x e. y}
Theorem	eluni 2502	Membership in class union.
|- (A e. U.B <-> E.x(A e. x /\ x e. B))
Theorem	eluni2 2503	Membership in class union. Restricted quantifier version.
|- (A e. U.B <-> E.x e. B A e. x)
Theorem	elunii 2504	Membership in class union.
|- ((A e. B /\ B e. C) -> A e. U.C)
Theorem	hbuni 2505	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 2506	Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.
|- (A = B -> U.A = U.B)
Theorem	unieqi 2507	Inference of equality of two class unions.
|- A = B   =>   |- U.A = U.B
Theorem	unieqd 2508	Deduction of equality of two class unions.
|- (ph -> A = B)   =>   |- (ph -> U.A = U.B)
Theorem	eluniab 2509	Membership in union of a class abstraction.
|- (A e. U.{x | ph} <-> E.x(A e. x /\ ph))
Theorem	elunirab 2510	Membership in union of a class abstraction.
|- (A e. U.{x e. B | ph} <-> E.x e. B (A e. x /\ ph))
Theorem	unipr 2511	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 2512	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 2513	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|- A e. V   =>   |- U.{A} = A
Theorem	unisng 2514	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|- (A e. B -> U.{A} = A)
Theorem	uniun 2515	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 2516	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 2517	Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.
|- (A (_ B -> U.A (_ U.B)
Theorem	ssuni 2518	Subclass relationship for class union.
|- ((A (_ B /\ B e. C) -> A (_ U.C)
Theorem	uni0b 2519	The union of a set is empty iff the set is included in the singleton of the empty set.
|- (U.A = (/) <-> A (_ {(/)})
Theorem	uni0c 2520	The union of a set is empty iff all of its members are empty.
|- (U.A = (/) <-> A.x e. A x = (/))
Theorem	uni0 2521	The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 2706 by Eric Schmidt, 4-Apr-2007.)
|- U.(/) = (/)
Theorem	elssuni 2522	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 2523	Condition turning a subclass relationship for union into an equality.
|- ((U.A (_ B /\ B e. A) -> U.A = B)
Theorem	unissb 2524	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 2525	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 2591 for a generalization to indexed unions.
|- (A.x e. A E.y e. B x (_ y -> U.A (_ U.B)
Theorem	unidif 2526	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 2527	Relationship implying union.
|- ((A e. B /\ A.x e. B x (_ A) -> A = U.B)
Theorem	unimax 2528	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 2529	Extend class notation to include the intersection of a class (read: 'intersect A ').
class |^|A
Definition	df-int 2530	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 2531	Alternate definition of class intersection.
|- |^|A = {x | A.y e. A x e. y}
Theorem	inteq 2532	Equality law for intersection.
|- (A = B -> |^|A = |^|B)
Theorem	inteqi 2533	Equality inference for class intersection.
|- A = B   =>   |- |^|A = |^|B
Theorem	inteqd 2534	Equality deduction for class intersection.
|- (ph -> A = B)   =>   |- (ph -> |^|A = |^|B)
Theorem	elint 2535	Membership in class intersection.
|- A e. V   =>   |- (A e. |^|B <-> A.x(x e. B -> A e. x))
Theorem	elint2 2536	Membership in class intersection.
|- A e. V   =>   |- (A e. |^|B <-> A.x e. B A e. x)
Theorem	elintg 2537	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 2538	Membership in class intersection.
|- (A e. |^|B -> (C e. B -> A e. C))
Theorem	hbint 2539	Bound-variable hypothesis builder for intersection.
|- (y e. A -> A.x y e. A)   =>   |- (y e. |^|A -> A.x y e. |^|A)
Theorem	elintab 2540	Membership in the intersection of a class abstraction.
|- A e. V   =>   |- (A e. |^|{x | ph} <-> A.x(ph -> A e. x))
Theorem	elintrab 2541	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 2542	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 2543	The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44.
|- |^|(/) = V
Theorem	intss1 2544	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 2545	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 2546	Subclass of the intersection of a class abstraction.
|- (A (_ |^|{x | ph} <-> A.x(ph -> A (_ x))
Theorem	ssintub 2547	Subclass of a least upper bound.
|- A (_ |^|{x e. B | A (_ x}
Theorem	ssmin 2548	Subclass of the minimum value of class of supersets.
|- A (_ |^|{x | (A (_ x /\ ph)}
Theorem	intmin 2549	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 2550	Intersection of subclasses.
|- (A (_ B -> |^|B (_ |^|A)
Theorem	intssuni 2551	The intersection of a nonempty set is a subclass of its union.
|- (A =/= (/) -> |^|A (_ U.A)
Theorem	intssuni2 2552	Subclass relationship for intersection and union.
|- ((A (_ B /\ A =/= (/)) -> |^|A (_ U.B)
Theorem	intmin2 2553	Any set is the smallest of all sets that include it.
|- A e. V   =>   |- |^|{x | A (_ x} = A
Theorem	intmin3 2554	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 2555	Elimination of a conjunct in a class intersection.
|- (A (_ |^|{x | ph} -> |^|{x | (A (_ x /\ ph)} = |^|{x | ph})
Theorem	intab 2556	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 3865 or abexssex 3866 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 2557	The intersection of a class containing the empty set is empty.
|- ((/) e. A -> |^|A = (/))
Theorem	intun 2558	The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42.
|- |^|(A u. B) = (|^|A i^i |^|B)
Theorem	intpr 2559	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 2560	The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41.
|- A e. V   =>   |- |^|{A} = A
Theorem	intunsn 2561	Theorem joining a singleton to an intersection.
|- B e. V   =>   |- |^|(A u. {B}) = (|^|A i^i B)
Indexed union and intersection
Syntax	ciun 2562	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 2499. 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 2563	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 2529. 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 2564	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 2583. Theorem uniiun 2597 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 3859 and funiunfv 3860 are useful when B is a function.
|- U_x e. A B = {y | E.x e. A y e. B}
Definition	df-iin 2565	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 2564. An alternate definition tying indexed intersection to ordinary intersection is dfiun2 2583. Theorem intiin 2598 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 2566	Membership in indexed union.
|- (A e. U_x e. B C <-> E.x e. B A e. C)
Theorem	eliin 2567	Membership in indexed intersection.
|- (A e. D -> (A e. |^|_x e. B C <-> A.x e. B A e. C))
Theorem	iunconst 2568	Indexed union of a constant class, i.e. where B does not depend on x.
|- (A =/= (/) -> U_x e. A B = B)
Theorem	iuniin 2569	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 2570	Subclass theorem for indexed union.
|- (A (_ B -> U_x e. A C (_ U_x e. B C)
Theorem	iuneq1 2571	Equality theorem for indexed union.
|- (A = B -> U_x e. A C = U_x e. B C)
Theorem	iineq1 2572	Equality theorem for restricted existential quantifier.
|- (A = B -> |^|_x e. A C = |^|_x e. B C)
Theorem	ss2iun 2573	Subclass theorem for indexed union.
|- (A.x e. A B (_ C -> U_x e. A B (_ U_x e. A C)
Theorem	iuneq2 2574	Equality theorem for indexed union.
|- (A.x e. A B = C -> U_x e. A B = U_x e. A C)
Theorem	iineq2