mmtheorems26 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10688

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 3872 or abexssex 3873 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 3866 and funiunfv 3867 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