mmtheorems27 - 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 10601-10700 108 10701-10783

Statement List for Metamath Proof Explorer - 2601-2700 - Page 27 of 108

Type	Label	Description
Statement
Theorem	uniiun 2601	Class union in terms of indexed union. Definition of [Stoll] p. 43.
|- U.A = U_x e. A x
Theorem	intiin 2602	Class intersection in terms of indexed intersection. Definition of [Stoll] p. 44.
|- |^|A = |^|_x e. A x
Theorem	iunid 2603	An indexed union of singletons recovers the index set.
|- U_x e. A {x} = A
Theorem	iun0 2604	An indexed union of the empty set is empty.
|- U_x e. A (/) = (/)
Theorem	0iun 2605	An empty indexed union is empty.
|- U_x e. (/) A = (/)
Theorem	0iin 2606	An empty indexed intersection is the universal class.
|- |^|_x e. (/) A = V
Theorem	iunn0 2607	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 2608	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 2601 to recover Enderton's theorem.
|- U_x e. A (B i^i C) = (B i^i U_x e. A C)
Theorem	iinun2 2609	Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 2602 to recover Enderton's theorem.
|- |^|_x e. A (B u. C) = (B u. |^|_x e. A C)
Theorem	iundif2 2610	Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 2602 to recover Enderton's theorem.
|- U_x e. A (B \ C) = (B \ |^|_x e. A C)
Theorem	iindif2 2611	Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 2601 to recover Enderton's theorem.
|- (A =/= (/) -> |^|_x e. A (B \ C) = (B \ U_x e. A C))
Theorem	iunxsn 2612	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 2613	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 2614	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 2615	A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33.
|- (A u. |^|B) = |^|_x e. B (A u. x)
Theorem	iununi 2616	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 2617	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 2618	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 2619	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 5293.)
wff ARB
Definition	df-br 2620	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 2621	Equality theorem for binary relations.
|- (R = S -> (ARB <-> ASB))
Theorem	breq1 2622	Equality theorem for a binary relation.
|- (A = B -> (ARC <-> BRC))
Theorem	breq2 2623	Equality theorem for a binary relation.
|- (A = B -> (CRA <-> CRB))
Theorem	breq12 2624	Equality theorem for a binary relation.
|- ((A = B /\ C = D) -> (ARC <-> BRD))
Theorem	breqi 2625	Equality inference for binary relations.
|- R = S   =>   |- (ARB <-> ASB)
Theorem	breq1i 2626	Equality inference for a binary relation.
|- A = B   =>   |- (ARC <-> BRC)
Theorem	breq2i 2627	Equality inference for a binary relation.
|- A = B   =>   |- (CRA <-> CRB)
Theorem	breq12i 2628	Equality inference for a binary relation. (The proof was shortened by Eric Schmidt, 4-Apr-2007.)
|- A = B   &   |- C = D   =>   |- (ARC <-> BRD)
Theorem	breq1d 2629	Equality deduction for a binary relation.
|- (ph -> A = B)   =>   |- (ph -> (ARC <-> BRC))
Theorem	breq2d 2630	Equality deduction for a binary relation.
|- (ph -> A = B)   =>   |- (ph -> (CRA <-> CRB))
Theorem	breq12d 2631	Equality deduction for a binary relation.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (ARC <-> BRD))
Theorem	breqan12d 2632	Equality deduction for a binary relation.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ph /\ ps) -> (ARC <-> BRD))
Theorem	breqan12rd 2633	Equality deduction for a binary relation.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ps /\ ph) -> (ARC <-> BRD))
Theorem	eqbrtr 2634	Substitution of equal classes into a binary relation.
|- A = B   &   |- BRC   =>   |- ARC
Theorem	eqbrtrd 2635	Substitution of equal classes into a binary relation.
|- (ph -> A = B)   &   |- (ph -> BRC)   =>   |- (ph -> ARC)
Theorem	eqbrtrr 2636	Substitution of equal classes into a binary relation.
|- A = B   &   |- ARC   =>   |- BRC
Theorem	eqbrtrrd 2637	Substitution of equal classes into a binary relation.
|- (ph -> A = B)   &   |- (ph -> ARC)   =>   |- (ph -> BRC)
Theorem	breqtr 2638	Substitution of equal classes into a binary relation.
|- ARB   &   |- B = C   =>   |- ARC
Theorem	breqtrd 2639	Substitution of equal classes into a binary relation.
|- (ph -> ARB)   &   |- (ph -> B = C)   =>   |- (ph -> ARC)
Theorem	breqtrr 2640	Substitution of equal classes into a binary relation.
|- ARB   &   |- C = B   =>   |- ARC
Theorem	breqtrrd 2641	Substitution of equal classes into a binary relation.
|- (ph -> ARB)   &   |- (ph -> C = B)   =>   |- (ph -> ARC)
Theorem	3brtr3 2642	Substitution of equality into both sides of a binary relation.
|- ARB   &   |- A = C   &   |- B = D   =>   |- CRD
Theorem	3brtr4 2643	Substitution of equality into both sides of a binary relation.
|- ARB   &   |- C = A   &   |- D = B   =>   |- CRD
Theorem	3brtr3d 2644	Substitution of equality into both sides of a binary relation.
|- (ph -> ARB)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> CRD)
Theorem	3brtr4d 2645	Substitution of equality into both sides of a binary relation.
|- (ph -> ARB)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> CRD)
Theorem	3brtr3g 2646	Substitution of equality into both sides of a binary relation.
|- (ph -> ARB)   &   |- A = C   &   |- B = D   =>   |- (ph -> CRD)
Theorem	3brtr4g 2647	Substitution of equality into both sides of a binary relation.
|- (ph -> ARB)   &   |- C = A   &   |- D = B   =>   |- (ph -> CRD)
Theorem	syl5eqbr 2648	A chained equality inference for a binary relation.
|- (ph -> ARB)   &   |- C = A   =>   |- (ph -> CRB)
Theorem	syl5eqbrr 2649	A chained equality inference for a binary relation.
|- (ph -> ARB)   &   |- A = C   =>   |- (ph -> CRB)
Theorem	syl5breq 2650	A chained equality inference for a binary relation.
|- (ph -> A = B)   &   |- CRA   =>   |- (ph -> CRB)
Theorem	syl5breqr 2651	A chained equality inference for a binary relation.
|- (ph -> B = A)   &   |- CRA   =>   |- (ph -> CRB)
Theorem	syl6eqbr 2652	A chained equality inference for a binary relation.
|- (ph -> A = B)   &   |- BRC   =>   |- (ph -> ARC)
Theorem	syl6eqbrr 2653	A chained equality inference for a binary relation.
|- (ph -> B = A)   &   |- BRC   =>   |- (ph -> ARC)
Theorem	syl6breq 2654	A chained equality inference for a binary relation.
|- (ph -> ARB)   &   |- B = C   =>   |- (ph -> ARC)
Theorem	syl6breqr 2655	A chained equality inference for a binary relation.
|- (ph -> ARB)   &   |- C = B   =>   |- (ph -> ARC)
Theorem	ssbrd 2656	Deduction from a subclass relationship of binary relations.
|- (ph -> A (_ B)   =>   |- (ph -> (CAD -> CBD))
Theorem	ssbri 2657	Inference from a subclass relationship of binary relations.
|- A (_ B   =>   |- (CAD -> CBD)
Theorem	hbbr 2658	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 2659	Deduction version of bound-variable hypothesis builder hbbr 2658.
|- (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 2660	Relationship between a binary relation and a class abstraction.
|- (xRy <-> x e. {z | zRy})
Theorem	brprc 2661	A property of proper class as the second argument of a binary relation.
|- (-. B e. V -> (ARB <-> ARA))
Theorem	sbcbrg 2662	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 2663	Move substitution in and out of a binary relation.
|- (A e. D -> ([A / x]BRC <-> [_A / x]_BR[_A / x]_C))
Theorem	sbcbr1g 2664	Move substitution in and out of a binary relation.
|- (A e. D -> ([A / x]BRC <-> [_A / x]_BRC))
Theorem	sbcbr2g 2665	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 2666	Extend class notation to include ordered-pair class abstraction (class builder).
class {<.x, y>. | ph}
Definition	df-opab 2667	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