mmtheorems27 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2601-2700 - Page 27 of 107

Type	Label	Description
Statement
Theorem	0iun 2601	An empty indexed union is empty.
|- U_x e. (/) A = (/)
Theorem	0iin 2602	An empty indexed intersection is the universal class.
|- |^|_x e. (/) A = V
Theorem	iunn0 2603	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 2604	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 2597 to recover Enderton's theorem.
|- U_x e. A (B i^i C) = (B i^i U_x e. A C)
Theorem	iinun2 2605	Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 2598 to recover Enderton's theorem.
|- |^|_x e. A (B u. C) = (B u. |^|_x e. A C)
Theorem	iundif2 2606	Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 2598 to recover Enderton's theorem.
|- U_x e. A (B \ C) = (B \ |^|_x e. A C)
Theorem	iindif2 2607	Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 2597 to recover Enderton's theorem.
|- (A =/= (/) -> |^|_x e. A (B \ C) = (B \ U_x e. A C))
Theorem	iunxsn 2608	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 2609	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 2610	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 2611	A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33.
|- (A u. |^|B) = |^|_x e. B (A u. x)
Theorem	iununi 2612	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	iinpw 2613	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 2614	Inclusion of an indexed intersection in the power class of a union. Part of Exercise 24(b) of [Enderton] p. 33.
|- U_x e. A P~x (_ P~U.A
Binary relations
Syntax	wbr 2615	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 5283.)
wff ARB
Definition	df-br 2616	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 2617	Equality theorem for binary relations.
|- (R = S -> (ARB <-> ASB))
Theorem	breq1 2618	Equality theorem for a binary relation.
|- (A = B -> (ARC <-> BRC))
Theorem	breq2 2619	Equality theorem for a binary relation.
|- (A = B -> (CRA <-> CRB))
Theorem	breq12 2620	Equality theorem for a binary relation.
|- ((A = B /\ C = D) -> (ARC <-> BRD))
Theorem	breqi 2621	Equality inference for binary relations.
|- R = S   =>   |- (ARB <-> ASB)
Theorem	breq1i 2622	Equality inference for a binary relation.
|- A = B   =>   |- (ARC <-> BRC)
Theorem	breq2i 2623	Equality inference for a binary relation.
|- A = B   =>   |- (CRA <-> CRB)
Theorem	breq12i 2624	Equality inference for a binary relation. (The proof was shortened by Eric Schmidt, 4-Apr-2007.)
|- A = B   &   |- C = D   =>   |- (ARC <-> BRD)
Theorem	breq1d 2625	Equality deduction for a binary relation.
|- (ph -> A = B)   =>   |- (ph -> (ARC <-> BRC))
Theorem	breq2d 2626	Equality deduction for a binary relation.
|- (ph -> A = B)   =>   |- (ph -> (CRA <-> CRB))
Theorem	breq12d 2627	Equality deduction for a binary relation.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (ARC <-> BRD))
Theorem	breqan12d 2628	Equality deduction for a binary relation.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ph /\ ps) -> (ARC <-> BRD))
Theorem	breqan12rd 2629	Equality deduction for a binary relation.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ps /\ ph) -> (ARC <-> BRD))
Theorem	eqbrtr 2630	Substitution of equal classes into a binary relation.
|- A = B   &   |- BRC   =>   |- ARC
Theorem	eqbrtrd 2631	Substitution of equal classes into a binary relation.
|- (ph -> A = B)   &   |- (ph -> BRC)   =>   |- (ph -> ARC)
Theorem	eqbrtrr 2632	Substitution of equal classes into a binary relation.
|- A = B   &   |- ARC   =>   |- BRC
Theorem	eqbrtrrd 2633	Substitution of equal classes into a binary relation.
|- (ph -> A = B)   &   |- (ph -> ARC)   =>   |- (ph -> BRC)
Theorem	breqtr 2634	Substitution of equal classes into a binary relation.
|- ARB   &   |- B = C   =>   |- ARC
Theorem	breqtrd 2635	Substitution of equal classes into a binary relation.
|- (ph -> ARB)   &   |- (ph -> B = C)   =>   |- (ph -> ARC)
Theorem	breqtrr 2636	Substitution of equal classes into a binary relation.
|- ARB   &   |- C = B   =>   |- ARC
Theorem	breqtrrd 2637	Substitution of equal classes into a binary relation.
|- (ph -> ARB)   &   |- (ph -> C = B)   =>   |- (ph -> ARC)
Theorem	3brtr3 2638	Substitution of equality into both sides of a binary relation.
|- ARB   &   |- A = C   &   |- B = D   =>   |- CRD
Theorem	3brtr4 2639	Substitution of equality into both sides of a binary relation.
|- ARB   &   |- C = A   &   |- D = B   =>   |- CRD
Theorem	3brtr3d 2640	Substitution of equality into both sides of a binary relation.
|- (ph -> ARB)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> CRD)
Theorem	3brtr4d 2641	Substitution of equality into both sides of a binary relation.
|- (ph -> ARB)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> CRD)
Theorem	3brtr3g 2642	Substitution of equality into both sides of a binary relation.
|- (ph -> ARB)   &   |- A = C   &   |- B = D   =>   |- (ph -> CRD)
Theorem	3brtr4g 2643	Substitution of equality into both sides of a binary relation.
|- (ph -> ARB)   &   |- C = A   &   |- D = B   =>   |- (ph -> CRD)
Theorem	syl5eqbr 2644	A chained equality inference for a binary relation.
|- (ph -> ARB)   &   |- C = A   =>   |- (ph -> CRB)
Theorem	syl5eqbrr 2645	A chained equality inference for a binary relation.
|- (ph -> ARB)   &   |- A = C   =>   |- (ph -> CRB)
Theorem	syl5breq 2646	A chained equality inference for a binary relation.
|- (ph -> A = B)   &   |- CRA   =>   |- (ph -> CRB)
Theorem	syl5breqr 2647	A chained equality inference for a binary relation.
|- (ph -> B = A)   &   |- CRA   =>   |- (ph -> CRB)
Theorem	syl6eqbr 2648	A chained equality inference for a binary relation.
|- (ph -> A = B)   &   |- BRC   =>   |- (ph -> ARC)
Theorem	syl6eqbrr 2649	A chained equality inference for a binary relation.
|- (ph -> B = A)   &   |- BRC   =>   |- (ph -> ARC)
Theorem	syl6breq 2650	A chained equality inference for a binary relation.
|- (ph -> ARB)   &   |- B = C   =>   |- (ph -> ARC)
Theorem	syl6breqr 2651	A chained equality inference for a binary relation.
|- (ph -> ARB)   &   |- C = B   =>   |- (ph -> ARC)
Theorem	ssbrd 2652	Deduction from a subclass relationship of binary relations.
|- (ph -> A (_ B)   =>   |- (ph -> (CAD -> CBD))
Theorem	ssbri 2653	Inference from a subclass relationship of binary relations.
|- A (_ B   =>   |- (CAD -> CBD)
Theorem	hbbr 2654	Bound-variable hypothesis builder for binary relation.
|- (y e. A -> A.x y e. A)   &   |- (y e. R -> A.x y e. R)   &   |- (y e. B -> A.x y e. B)   =>   |- (ARB -> A.x ARB)
Theorem	hbbrd 2655	Deduction version of bound-variable hypothesis builder hbbr 2654.
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   &   |- (ph -> (y e. R -> A.x y e. R))   &   |- (ph -> (y e. B -> A.x y e. B))   =>   |- (ph -> (ARB -> A.x ARB))
Theorem	brab1 2656	Relationship between a binary relation and a class abstraction.
|- (xRy <-> x e. {z | zRy})
Theorem	brprc 2657	A property of proper class as the second argument of a binary relation.
|- (-. B e. V -> (ARB <-> ARA))
Theorem	sbcbrg 2658	Move substitution in and out of a binary relation.
|- (A e. D -> ([A / x]BRC <-> [_A / x]_B[_A / x]_R[_A / x]_C))
Theorem	sbcbr12g 2659	Move substitution in and out of a binary relation.
|- (A e. D -> ([A / x]BRC <-> [_A / x]_BR[_A / x]_C))
Theorem	sbcbr1g 2660	Move substitution in and out of a binary relation.
|- (A e. D -> ([A / x]BRC <-> [_A / x]_BRC))
Theorem	sbcbr2g 2661	Move substitution in and out of a binary relation.
|- (A e. D -> ([A / x]BRC <-> BR[_A / x]_C))
Ordered-pair class abstractions (class builders)
Syntax	copab 2662	Extend class notation to include ordered-pair class abstraction (class builder).
class {<.x, y>. | ph}
Definition	df-opab 2663	Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually x and y are distinct, although the definition doesn't strictly require it (see dfid2 2835 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. An alternate definition using no existential quantifiers is shown by dfopab2 4112.
|- {<.x, y>. | ph} = {z | E.xE.y(z = <.x, y>. /\ ph)}
Theorem	opabss 2664	The collection of ordered pairs in a class is a subclass of it.
|- {<.x, y>. | xRy} (_ R
Theorem	opabbid 2665	Equivalent wff's yield equal ordered-pair class abstractions (deduction rule).