mmtheorems26 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2501-2600 - Page 26 of 108

Type	Label	Description
Statement
Theorem	hbopd 2501	Deduction version of bound-variable hypothesis builder hbop 2500.
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   &   |- (ph -> (y e. B -> A.x y e. B))   =>   |- (ph -> (y e. <.A, B>. -> A.x y e. <.A, B>.))
Theorem	opprc1 2502	Expansion of an ordered pair when the first member is a proper class. See also opprc1b 2802, opprc2 2503, opprc3 2803.
|- (-. A e. V -> <.A, B>. = {(/), {B}})
Theorem	opprc2 2503	A property of an ordered pair of proper classes (due to our particular definition of ordered pair).
|- (-. B e. V -> <.A, B>. = <.A, A>.)
Theorem	pwsn 2504	The power set of a singleton.
|- P~{A} = {(/), {A}}
Theorem	pwsnALT 2505	The power set of a singleton (direct proof).
|- P~{A} = {(/), {A}}
Theorem	pwv 2506	The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235.
|- P~V = V
The union of a class
Syntax	cuni 2507	Extend class notation to include the union of a class (read: 'union A ')
class U.A
Definition	df-uni 2508	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 2509	Alternate definition of class union.
|- U.A = {x | E.y e. A x e. y}
Theorem	eluni 2510	Membership in class union.
|- (A e. U.B <-> E.x(A e. x /\ x e. B))
Theorem	eluni2 2511	Membership in class union. Restricted quantifier version.
|- (A e. U.B <-> E.x e. B A e. x)
Theorem	elunii 2512	Membership in class union.
|- ((A e. B /\ B e. C) -> A e. U.C)
Theorem	hbuni 2513	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 2514	Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.
|- (A = B -> U.A = U.B)
Theorem	unieqi 2515	Inference of equality of two class unions.
|- A = B   =>   |- U.A = U.B
Theorem	unieqd 2516	Deduction of equality of two class unions.
|- (ph -> A = B)   =>   |- (ph -> U.A = U.B)
Theorem	eluniab 2517	Membership in union of a class abstraction.
|- (A e. U.{x | ph} <-> E.x(A e. x /\ ph))
Theorem	elunirab 2518	Membership in union of a class abstraction.
|- (A e. U.{x e. B | ph} <-> E.x e. B (A e. x /\ ph))
Theorem	unipr 2519	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 2520	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 2521	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|- A e. V   =>   |- U.{A} = A
Theorem	unisng 2522	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
|- (A e. B -> U.{A} = A)
Theorem	uniun 2523	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 2524	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 2525	Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.
|- (A (_ B -> U.A (_ U.B)
Theorem	ssuni 2526	Subclass relationship for class union.
|- ((A (_ B /\ B e. C) -> A (_ U.C)
Theorem	uni0b 2527	The union of a set is empty iff the set is included in the singleton of the empty set.
|- (U.A = (/) <-> A (_ {(/)})
Theorem	uni0c 2528	The union of a set is empty iff all of its members are empty.
|- (U.A = (/) <-> A.x e. A x = (/))
Theorem	uni0 2529	The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 2715 by Eric Schmidt, 4-Apr-2007.)
|- U.(/) = (/)
Theorem	elssuni 2530	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 2531	Condition turning a subclass relationship for union into an equality.
|- ((U.A (_ B /\ B e. A) -> U.A = B)
Theorem	unissb 2532	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 2533	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 2599 for a generalization to indexed unions.
|- (A.x e. A E.y e. B x (_ y -> U.A (_ U.B)
Theorem	unidif 2534	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 2535	Relationship implying union.
|- ((A e. B /\ A.x e. B x (_ A) -> A = U.B)
Theorem	unimax 2536	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 2537	Extend class notation to include the intersection of a class (read: 'intersect A ').
class |^|A
Definition	df-int 2538	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 2539	Alternate definition of class intersection.
|- |^|A = {x | A.y e. A x e. y}
Theorem	inteq 2540	Equality law for intersection.
|- (A = B -> |^|A = |^|B)
Theorem	inteqi 2541	Equality inference for class intersection.
|- A = B   =>   |- |^|A = |^|B
Theorem	inteqd 2542	Equality deduction for class intersection.
|- (ph -> A = B)   =>   |- (ph -> |^|A = |^|B)
Theorem	elint 2543	Membership in class intersection.
|- A e. V   =>   |- (A e. |^|B <-> A.x(x e. B -> A e. x))
Theorem	elint2 2544	Membership in class intersection.
|- A e. V   =>   |- (A e. |^|B <-> A.x e. B A e. x)
Theorem	elintg 2545	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 2546	Membership in class intersection.
|- (A e. |^|B -> (C e. B -> A e. C))
Theorem	hbint 2547	Bound-variable hypothesis builder for intersection.
|- (y e. A -> A.x y e. A)   =>   |- (y e. |^|A -> A.x y e. |^|A)
Theorem	elintab 2548	Membership in the intersection of a class abstraction.
|- A e. V   =>   |- (A e. |^|{x | ph} <-> A.x(ph -> A e. x))
Theorem	elintrab 2549	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 2550	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 2551	The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44.
|- |^|(/) = V
Theorem	intss1 2552	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 2553	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 2554	Subclass of the intersection of a class abstraction.
|- (A (_ |^|{x | ph} <-> A.x(ph -> A (_ x))
Theorem	ssintub 2555	Subclass of a least upper bound.
|- A (_ |^|{x e. B | A (_ x}
Theorem	ssmin 2556	Subclass of the minimum value of class of supersets.
|- A (_ |^|{x | (A (_ x /\ ph)}
Theorem	intmin 2557	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 2558	Intersection of subclasses.
|- (A (_ B -> |^|B (_ |^|A)
Theorem	intssuni 2559	The intersection of a nonempty set is a subclass of its union.
|- (A =/= (/) -> |^|A (_ U.A)
Theorem	intssuni2 2560	Subclass relationship for intersection and union.
|- ((A (_ B /\ A =/= (/)) -> |^|A (_ U.B)
Theorem	intmin2 2561	Any set is the smallest of all sets that include it.
|- A e. V   =>   |- |^|{x | A (_ x} = A
Theorem	intmin3 2562	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 2563	Elimination of a conjunct in a class intersection.
|- (A (_ |^|{x | ph} -> |^|{x | (A (_ x /\ ph)} = |^|{x | ph})
Theorem	intab 2564	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 3877 or abexssex 3878 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 2565	The intersection of a class containing the empty set is empty.
|- ((/) e. A -> |^|A = (/))
Theorem	intun 2566	The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42.
|- |^|(A u. B) = (|^|A i^i |^|B)
Theorem	intpr 2567	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 2568	The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41.
|- A e. V   =>   |- |^|{A} = A
Theorem	intunsn 2569	Theorem joining a singleton to an intersection.
|- B e. V   =>   |- |^|(A u. {B}) = (|^|A i^i B)
Indexed union and intersection
Syntax	ciun 2570	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 2507. 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 2571	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 2537. 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 2572	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 2591. Theorem uniiun 2605 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 3871 and funiunfv 3872 are useful when B is a function.
|- U_x e. A B = {y | E.x e. A y e. B}
Definition	df-iin 2573	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 2572. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 2592. Theorem intiin 2606 provides a definition of ordinary intersection in terms of indexed intersection.
|- |^|_x e. A B = {y | A.x e. A y e. B}