mmtheorems27 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2601-2700 - Page 27 of 123

Type	Label	Description
Statement
Definition	df-int 2601	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 2602	Alternate definition of class intersection.
|- |^|A = {x | A.y e. A x e. y}
Theorem	inteq 2603	Equality law for intersection.
|- (A = B -> |^|A = |^|B)
Theorem	inteqi 2604	Equality inference for class intersection.
|- A = B   =>   |- |^|A = |^|B
Theorem	inteqd 2605	Equality deduction for class intersection.
|- (ph -> A = B)   =>   |- (ph -> |^|A = |^|B)
Theorem	elint 2606	Membership in class intersection.
|- A e. V   =>   |- (A e. |^|B <-> A.x(x e. B -> A e. x))
Theorem	elint2 2607	Membership in class intersection.
|- A e. V   =>   |- (A e. |^|B <-> A.x e. B A e. x)
Theorem	elintg 2608	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 2609	Membership in class intersection.
|- (A e. |^|B -> (C e. B -> A e. C))
Theorem	hbint 2610	Bound-variable hypothesis builder for intersection.
|- (y e. A -> A.x y e. A)   =>   |- (y e. |^|A -> A.x y e. |^|A)
Theorem	elintab 2611	Membership in the intersection of a class abstraction.
|- A e. V   =>   |- (A e. |^|{x | ph} <-> A.x(ph -> A e. x))
Theorem	elintrab 2612	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 2613	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 2614	The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44.
|- |^|(/) = V
Theorem	intss1 2615	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 2616	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 2617	Subclass of the intersection of a class abstraction.
|- (A (_ |^|{x | ph} <-> A.x(ph -> A (_ x))
Theorem	ssintub 2618	Subclass of a least upper bound.
|- A (_ |^|{x e. B | A (_ x}
Theorem	ssmin 2619	Subclass of the minimum value of class of supersets.
|- A (_ |^|{x | (A (_ x /\ ph)}
Theorem	intmin 2620	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 2621	Intersection of subclasses.
|- (A (_ B -> |^|B (_ |^|A)
Theorem	intssuni 2622	The intersection of a nonempty set is a subclass of its union.
|- (A =/= (/) -> |^|A (_ U.A)
Theorem	intssuni2 2623	Subclass relationship for intersection and union.
|- ((A (_ B /\ A =/= (/)) -> |^|A (_ U.B)
Theorem	intmin2 2624	Any set is the smallest of all sets that include it.
|- A e. V   =>   |- |^|{x | A (_ x} = A
Theorem	intmin3 2625	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 2626	Elimination of a conjunct in a class intersection.
|- (A (_ |^|{x | ph} -> |^|{x | (A (_ x /\ ph)} = |^|{x | ph})
Theorem	intab 2627	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 3985 or abexssex 3986 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 2628	The intersection of a class containing the empty set is empty.
|- ((/) e. A -> |^|A = (/))
Theorem	intun 2629	The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42.
|- |^|(A u. B) = (|^|A i^i |^|B)
Theorem	intpr 2630	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 2631	The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41.
|- A e. V   =>   |- |^|{A} = A
Theorem	intunsn 2632	Theorem joining a singleton to an intersection.
|- B e. V   =>   |- |^|(A u. {B}) = (|^|A i^i B)
Indexed union and intersection
Syntax	ciun 2633	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 2569. 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 2634	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 2600. 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 2635	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 2655. Theorem uniiun 2669 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 3979 and funiunfv 3980 are useful when B is a function.
|- U_x e. A B = {y | E.x e. A y e. B}
Definition	df-iin 2636	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 2635. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 2656. Theorem intiin 2670 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 2637	Membership in indexed union.
|- (A e. U_x e. B C <-> E.x e. B A e. C)
Theorem	eliin 2638	Membership in indexed intersection.
|- (A e. D -> (A e. |^|_x e. B C <-> A.x e. B A e. C))
Theorem	iuncom 2639	Commutation of indexed unions.
|- U_x e. A U_y e. B C = U_y e. B U_x e. A C
Theorem	iunconst 2640	Indexed union of a constant class, i.e. where B does not depend on x.
|- (A =/= (/) -> U_x e. A B = B)
Theorem	iuniin 2641	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 2642	Subclass theorem for indexed union.
|- (A (_ B -> U_x e. A C (_ U_x e. B C)
Theorem	iuneq1 2643	Equality theorem for indexed union.
|- (A = B -> U_x e. A C = U_x e. B C)
Theorem	iineq1 2644	Equality theorem for restricted existential quantifier.
|- (A = B -> |^|_x e. A C = |^|_x e. B C)
Theorem	ss2iun 2645	Subclass theorem for indexed union.
|- (A.x e. A B (_ C -> U_x e. A B (_ U_x e. A C)
Theorem	iuneq2 2646	Equality theorem for indexed union.
|- (A.x e. A B = C -> U_x e. A B = U_x e. A C)
Theorem	iineq2 2647	Equality theorem for indexed intersection.
|- (A.x e. A B = C -> |^|_x e. A B = |^|_x e. A C)
Theorem	iuneq2i 2648	Equality inference for indexed union.
|- (x e. A -> B = C)   =>   |- U_x e. A B = U_x e. A C
Theorem	iineq2i 2649	Equality inference for indexed intersection.
|- (x e. A -> B = C)   =>   |- |^|_x e. A B = |^|_x e. A C
Theorem	iuneq2dv 2650	Equality deduction for indexed union.
|- ((ph /\ x e. A) -> B = C)   =>   |- (ph -> U_x e. A B = U_x e. A C)
Theorem	iineq2dv 2651	Equality deduction for indexed intersection.
|- ((ph /\ x e. A) -> B = C)   =>   |- (ph -> |^|_x e. A B = |^|_x e. A C)
Theorem	hbiu1 2652	Bound-variable hypothesis builder for indexed union.
|- (y e. U_x e. A B -> A.x y e. U_x e. A B)
Theorem	hbii1 2653	Bound-variable hypothesis builder for indexed intersection.
|- (y e. |^|_x e. A B -> A.x y e. |^|_x e. A B)
Theorem	dfiun2g 2654	Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44.
|- (A.x e. A B e. C -> U_x e. A B = U.{y | E.x e. A y = B})
Theorem	dfiun2 2655	Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44.
|- B e. V   =>   |- U_x e. A B = U.{y | E.x e. A y = B}
Theorem	dfiin2 2656	Alternate definition of indexed intersection when B is a set. Definition 15(b) of [Suppes] p. 44.
|- B e. V   =>   |- |^|_x e. A B = |^|{y | E.x e. A y = B}
Theorem	cbviun 2657	Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis.
|- (z e. B -> A.y z e. B)   &   |- (z e. C -> A.x z e. C)   &   |- (x = y -> B = C)   =>   |- U_x e. A B = U_y e. A C
Theorem	cbviunv 2658	Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis.
|- (x = y -> B = C)   =>   |- U_x e. A B = U_y e. A C
Theorem	iunss 2659	Subset theorem for an indexed union.
|- (U_x e. A B (_ C <-> A.x e. A B (_ C)
Theorem	ssiun 2660	Subset implication for an indexed union.
|- (E.x e. A C (_ B -> C (_ U_x e. A B)
Theorem	ssiun2 2661	Identity law for subset of an indexed union.
|- (x e. A -> B (_ U_x e. A B)
Theorem	ssiun2s 2662	Subset relationship for an indexed union.
|- (x = C -> B = D)   =>   |- (C e. A -> D (_ U_x e. A B)
Theorem	iunss2 2663	A subclass condition on the members of two indexed classes C(x) and D(y) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 2596.
|- (A.x e. A E.y e. B C (_ D -> U_x e. A C (_ U_y e. B D)
Theorem	iunrab 2664	The indexed union of a restricted class abstraction.
|- U_x e. A {y e. B | ph} = {y e. B | E.x e. A ph}
Theorem	iunab 2665	The indexed union of a class abstraction.
|- U_x e. A {y | ph} = {y | E.x e. A ph}
Theorem	iunxdif2 2666	Indexed union with a class difference as its index.
|- (x = y -> C = D)   =>   |- (A.x e. A E.y e. (A \ B)C (_ D -> U_y e. (A \ B)D = U_x e. A C)
Theorem	ssiin 2667	Subset theorem for an indexed intersection.
|- (C (_ |^|_x e. A B <-> A.x e. A C (_ B)
Theorem	iinss 2668	Subset implication for an indexed intersection.
|- (E.x e. A B (_ C -> |^|_x e. A B (_ C)
Theorem	uniiun 2669	Class union in terms of indexed union. Definition of [Stoll] p. 43.
|- U.A = U_x e. A x
Theorem	intiin 2670	Class intersection in terms of indexed intersection. Definition of [Stoll] p. 44.
|- |^|A = |^|_x e. A x
Theorem	iunid 2671	An indexed union of singletons recovers the index set.
|- U_x e. A {x} = A
Theorem	iun0 2672	An indexed union of the empty set is empty.
|- U_x e. A (/) = (/)
Theorem	0iun 2673	An empty indexed union is empty.
|- U_x e. (/) A = (/)
Theorem	0iin 2674	An empty indexed intersection is the universal class.
|- |^|_x e. (/) A = V
Theorem	viin 2675	Indexed intersection with a universal index class. When A doesn't depend on x, this evaluates to A by 19.3 1067 and abid2 1623. When A = x, this evaluates to (/) by intiin 2670 and intv 2816.
|- |^|_x e. V A = {y | A.x y e. A}
Theorem	iunn0 2676	There is a non-empty class in an indexed collection B(x) iff the indexed union of them is non-empty.
|- (E.x e. A B =/= (/) <-> U_x e. A B =/= (/))
Theorem	iunin2 2677	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 2669 to recover Enderton's theorem.
|- U_x e. A (B i^i C) = (B i^i U_x e. A C)
Theorem	iinun2 2678	Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 2670 to recover Enderton's theorem.
|- |^|_x e. A (B u. C) = (B u. |^|_x e. A C)
Theorem	iundif2 2679	Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 2670 to recover Enderton's theorem.
|- U_x e. A (B \ C) = (B \ |^|_x e. A C)
Theorem	2iunin 2680	Rearrange indexed unions over intersection.
|- U_x e. A U_y e. B (C i^i D) = (U_x e. A C i^i U_y e. B D)
Theorem	iindif2 2681	Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 2669 to recover Enderton's theorem.
|- (A =/= (/) -> |^|_x e. A (B \ C) = (B \ U_x e. A C))
Theorem	iunxsn 2682	A singleton index picks out an instance of an indexed union's argument.
|- A e. V   &   |- (x = A -> B = C)   =>   |- U_x e. {A}B = C
Theorem	iunun 2683	Separate a union in an indexed union.
|- U_x e. A (B u. C) = (U_x e. A B u. U_x e. A C)
Theorem	iunxun 2684	Separate a union in the index of an indexed union.
|- U_x e. (A u. B)C = (U_x e. A C u. U_x e. B C)
Theorem	iinuni 2685	A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33.
|- (A u. |^|B) = |^|_x e. B (A u. x)
Theorem	iununi 2686	A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33.
|- ((B = (/) -> A = (/)) <-> (A u. U.B) = U_x e. B (A u. x))
Theorem	sspwuni 2687	Subclass relationship for power class and union.
|- (A (_ P~B <-> U.A (_ B)
Theorem	pwssb 2688	Two ways to express a collection of subclasses.
|- (A (_ P~B <-> A.x e. A x (_ B)
Theorem	elpwuni 2689	Relationship for power class and union.
|- (B e. A -> (A (_ P~B <-> U.A = B))
Theorem	iinpw 2690	The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33.
|- P~|^|A = |^|_x e. A P~x
Theorem	iunpwss 2691	Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33.
|- U_x e. A P~x (_ P~U.A
Binary relations
Syntax	wbr 2692	Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 5447.)
wff ARB
Definition	df-br 2693	Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. Class R normally denotes a relation such as "<" that compares two classes A and B, which might be numbers such as 1 and 2. This definition is well-defined, although not very meaningful, when classes A and/or B are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when R is a proper class.
|- (ARB <-> <.A, B>. e. R)
Theorem	breq 2694	Equality theorem for binary relations.
|- (R = S -> (ARB <-> ASB))
Theorem	breq1 2695	Equality theorem for a binary relation.
|- (A = B -> (ARC <-> BRC))
Theorem	breq2 2696	Equality theorem for a binary relation.
|- (A = B -> (CRA <-> CRB))
Theorem	breq12 2697	Equality theorem for a binary relation.
|- ((A = B /\ C = D) -> (ARC <-> BRD))
Theorem	breqi 2698	Equality inference for binary relations.
|- R = S   =>   |- (ARB <-> ASB)
Theorem	breq1i 2699	Equality inference for a binary relation.
|- A = B   =>   |- (ARC <-> BRC)
Theorem	breq2i 2700	Equality inference for a binary relation.
|- A = B   =>   |- (CRA <-> CRB)