mmtheorems25 - 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 10601-10691

Statement List for Metamath Proof Explorer - 2401-2500 - Page 25 of 107

Type	Label	Description
Statement
Theorem	elpw 2401	Membership in a power class. Theorem 86 of [Suppes] p. 47.
|- A e. V   =>   |- (A e. P~B <-> A (_ B)
Theorem	elpwg 2402	Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 2723.
|- (A e. C -> (A e. P~B <-> A (_ B))
Theorem	elpwi 2403	Subset relation implied by membership in a power class.
|- (A e. P~B -> A (_ B)
Theorem	hbpw 2404	Bound-variable hypothesis builder for power class.
|- (y e. A -> A.x y e. A)   =>   |- (y e. P~A -> A.x y e. P~A)
Theorem	pwid 2405	A set is a member of its power class. Theorem 87 of [Suppes] p. 47.
|- A e. V   =>   |- A e. P~A
Unordered and ordered pairs
Syntax	csn 2406	Extend class notation to include singleton.
class {A}
Syntax	cpr 2407	Extend class notation to include unordered pair.
class {A, B}
Syntax	cop 2408	Extend class notation to include ordered pair.
class <.A, B>.
Definition	df-sn 2409	Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of V, although it is not very meaningful in this case. For an alternate definition see dfsn2 2417.
|- {A} = {x | x = A}
Definition	df-pr 2410	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For a more traditional definition, but requiring a dummy variable, see dfpr2 2419.
|- {A, B} = ({A} u. {B})
Syntax	ctp 2411	Extend class notation to include unordered triplet.
class {A, B, C}
Definition	df-tp 2412	Define unordered triple of classes. Definition of [Enderton] p. 19.
|- {A, B, C} = ({A, B} u. {C})
Definition	df-op 2413	Kuratowski's ordered pair definition. Definition 9.1 of [Quine] p. 58. For proper classes it is not meaningful but is well-defined and we allow it for convenience (see opprc1 2495, opprc1b 2792, opprc2 2496, and opprc3 2793). For the justifying theorem (for sets) see opth 2783. There are other ways to define ordered pairs; the basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition <.A, B>.2 = {{{A}, (/)}, {{B}}}, justified by opthwiener 2803, which was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition <.A, B>.3 = {A, {A, B}} is justified by opthreg 4587, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is <.A, B>.4 = ((A X. {(/)}) u. (B X. {{(/)}})), justified by opthprc 3217. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opth 6612. Finally, an ordered pair of real numbers can be represented by a complex number as shown by cru 6683.
|- <.A, B>. = {{A}, {A, B}}
Theorem	sneq 2414	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15.
|- (A = B -> {A} = {B})
Theorem	sneqi 2415	Equality inference for singletons.
|- A = B   =>   |- {A} = {B}
Theorem	sneqd 2416	Equality deduction for singletons.
|- (ph -> A = B)   =>   |- (ph -> {A} = {B})
Theorem	dfsn2 2417	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15.
|- {A} = {A, A}
Theorem	elsn 2418	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15.
|- (x e. {A} <-> x = A)
Theorem	dfpr2 2419	Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15.
|- {A, B} = {x | (x = A \/ x = B)}
Theorem	elprg 2420	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized.
|- (A e. D -> (A e. {B, C} <-> (A = B \/ A = C)))
Theorem	elpr 2421	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15.
|- A e. V   =>   |- (A e. {B, C} <-> (A = B \/ A = C))
Theorem	elpr2 2422	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15.
|- B e. V   &   |- C e. V   =>   |- (A e. {B, C} <-> (A = B \/ A = C))
Theorem	hbpr 2423	Bound-variable hypothesis builder for unordered pairs.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. {A, B} -> A.x y e. {A, B})
Theorem	ifpr 2424	Membership of a conditional operator in an unordered pair.
|- ((A e. C /\ B e. D) -> if(ph, A, B) e. {A, B})
Theorem	ralpr 2425	Convert a quantification over a pair to a conjunction.
|- A e. V   &   |- B e. V   =>   |- (A.x e. {A, B}ph <-> ([A / x]ph /\ [B / x]ph))
Theorem	rexpr 2426	Convert an existential quantification over a pair to a disjunction.
|- A e. V   &   |- B e. V   =>   |- (E.x e. {A, B}ph <-> ([A / x]ph \/ [B / x]ph))
Theorem	elsncg 2427	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized).
|- (A e. C -> (A e. {B} <-> A = B))
Theorem	elsnc 2428	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15.
|- A e. V   =>   |- (A e. {B} <-> A = B)
Theorem	elsni 2429	There is only one element in a singleton.
|- (A e. {B} -> A = B)
Theorem	snidg 2430	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49.
|- (A e. B -> A e. {A})
Theorem	snidb 2431	A class is a set iff it is a member of its singleton.
|- (A e. V <-> A e. {A})
Theorem	snid 2432	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49.
|- A e. V   =>   |- A e. {A}
Theorem	elsnc2g 2433	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set.
|- (B e. C -> (A e. {B} <-> A = B))
Theorem	elsnc2 2434	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set.
|- B e. V   =>   |- (A e. {B} <-> A = B)
Theorem	hbsn 2435	Bound-variable hypothesis builder for singletons.
|- (y e. A -> A.x y e. A)   =>   |- (y e. {A} -> A.x y e. {A})
Theorem	eltp 2436	A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17.
|- A e. V   =>   |- (A e. {B, C, D} <-> (A = B \/ A = C \/ A = D))
Theorem	dftp2 2437	Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16.
|- {A, B, C} = {x | (x = A \/ x = B \/ x = C)}
Theorem	disjsn 2438	Intersection with the singleton of a non-member is disjoint.
|- ((A i^i {B}) = (/) <-> -. B e. A)
Theorem	disjsn2 2439	Intersection of distinct singletons is disjoint.
|- (A =/= B -> ({A} i^i {B}) = (/))
Theorem	snprc 2440	The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48.
|- (-. A e. V <-> {A} = (/))
Theorem	r19.12sn 2441	Special case of r19.12 1738 where its converse holds.
|- A e. V   =>   |- (E.x e. {A}A.y e. B ph <-> A.y e. B E.x e. {A}ph)
Theorem	rabsn 2442	Condition where a restricted class abstraction is a singleton.
|- (B e. A -> {x e. A | x = B} = {B})
Theorem	eusn 2443	Another way to express existential uniqueness of a wff: its class abstraction is a singleton.
|- (E!xph <-> E.x{x | ph} = {x})
Theorem	prcom 2444	Commutative law for unordered pairs.
|- {A, B} = {B, A}
Theorem	preq1 2445	An equality theorem for unordered pairs.
|- (A = B -> {A, C} = {B, C})
Theorem	preq2 2446	An equality theorem for unordered pairs.
|- (A = B -> {C, A} = {C, B})
Theorem	pri1 2447	One of the two elements of an unordered pair. Part of Theorem 7.6 of [Quine] p. 49.
|- A e. V   =>   |- A e. {A, B}
Theorem	pri2 2448	One of the two elements of an unordered pair. Part of Theorem 7.6 of [Quine] p. 49.
|- B e. V   =>   |- B e. {A, B}
Theorem	prprc1 2449	A proper class vanishes in an unordered pair.
|- (-. A e. V -> {A, B} = {B})
Theorem	prprc2 2450	A proper class vanishes in an unordered pair.
|- (-. B e. V -> {A, B} = {A})
Theorem	prprc 2451	An unordered pair containing two proper classes is the empty set.
|- ((-. A e. V /\ -. B e. V) -> {A, B} = (/))
Theorem	tpi1 2452	One of the three elements of an unordered triple.
|- A e. V   =>   |- A e. {A, B, C}
Theorem	tpi2 2453	One of the three elements of an unordered triple.
|- B e. V   =>   |- B e. {A, B, C}
Theorem	tpi3 2454	One of the three elements of an unordered triple.
|- C e. V   =>   |- C e. {A, B, C}
Theorem	snnz 2455	The singleton of a set is not empty.
|- A e. V   =>   |- {A} =/= (/)
Theorem	prnz 2456	A pair containing a set is not empty.
|- A e. V   =>   |- {A, B} =/= (/)
Theorem	tpnz 2457	A triplet containing a set is not empty.
|- A e. V   =>   |- {A, B, C} =/= (/)
Theorem	snss 2458	The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49.
|- A e. V   =>   |- (A e. B <-> {A} (_ B)
Theorem	eldifsn 2459	Membership in a set with an element removed.
|- (A e. (B \ {C}) <-> (A e. B /\ A =/= C))
Theorem	snssg 2460	The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49.
|- (A e. C -> (A e. B <-> {A} (_ B))
Theorem	difsn 2461	An element not in a set can be removed without affecting the set.
|- (-. A e. B -> (B \ {A}) = B)
Theorem	difprsn 2462	Removal of a singleton from an unordered pair.
|- ({A, B} \ {A}) (_ {B}
Theorem	snssi 2463	The singleton of an element of a class is a subset of the class.
|- (A e. B -> {A} (_ B)
Theorem	difsnid 2464	If we remove a single element from a class then put it back in, we end up with the original class.
|- (B e. A -> ((A \ {B}) u. {B}) = A)
Theorem	pw0 2465	Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (The proof was shortened by Eric Schmidt, 4-Apr-2007.)
|- P~(/) = {(/)}
Theorem	pwpw0 2466	Compute the power set of the power set of the empty set. (See pw0 2465 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 2497, we have chosen to show a direct elementary proof.
|- P~{(/)} = {(/), {(/)}}
Theorem	snsspr 2467	A singleton is a subset of an unordered pair containing its member.
|- {A} (_ {A, B}
Theorem	prss 2468	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49.
|- A e. V   &   |- B e. V   =>   |- ((A e. C /\ B e. C) <-> {A, B} (_ C)
Theorem	prssg 2469	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49.
|- ((A e. R /\ B e. S) -> ((A e. C /\ B e. C) <-> {A, B} (_ C))
Theorem	sssn 2470	The subsets of a singleton.
|- (A (_ {B} <-> (A = (/) \/ A = {B}))
Theorem	eqsn 2471	Two ways to express that a nonempty set equals a singleton.
|- (A =/= (/) -> (A = {B} <-> A.x e. A x = B))
Theorem	sspr 2472	The subsets of a pair.
|- (A (_ {B, C} <-> ((A = (/) \/ A = {B}) \/ (A = {C} \/ A = {B, C})))
Theorem	tpss 2473	A triplet of elements of a class is a subset of the class.
|- A e. V   &   |- B