mmtheorems25 - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	elpwg 2401	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 2402	Subset relation implied by membership in a power class.
|- (A e. P~B -> A (_ B)
Theorem	hbpw 2403	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 2404	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 2405	Extend class notation to include singleton.
class {A}
Syntax	cpr 2406	Extend class notation to include unordered pair.
class {A, B}
Syntax	cop 2407	Extend class notation to include ordered pair.
class <.A, B>.
Definition	df-sn 2408	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 2416.
|- {A} = {x | x = A}
Definition	df-pr 2409	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For a more traditional definition, but requiring a dummy variable, see dfpr2 2418.
|- {A, B} = ({A} u. {B})
Syntax	ctp 2410	Extend class notation to include unordered triplet.
class {A, B, C}
Definition	df-tp 2411	Define unordered triple of classes. Definition of [Enderton] p. 19.
|- {A, B, C} = ({A, B} u. {C})
Definition	df-op 2412	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 2804, 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 4595, 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 3221. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opth 6615. Finally, an ordered pair of real numbers can be represented by a complex number as shown by cru 6686.
|- <.A, B>. = {{A}, {A, B}}
Theorem	sneq 2413	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15.
|- (A = B -> {A} = {B})
Theorem	sneqi 2414	Equality inference for singletons.
|- A = B   =>   |- {A} = {B}
Theorem	sneqd 2415	Equality deduction for singletons.
|- (ph -> A = B)   =>   |- (ph -> {A} = {B})
Theorem	dfsn2 2416	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15.
|- {A} = {A, A}
Theorem	elsn 2417	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15.
|- (x e. {A} <-> x = A)
Theorem	dfpr2 2418	Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15.
|- {A, B} = {x | (x = A \/ x = B)}
Theorem	elprg 2419	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 2420	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 2421	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 2422	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 2423	Membership of a conditional operator in an unordered pair.
|- ((A e. C /\ B e. D) -> if(ph, A, B) e. {A, B})
Theorem	ralpr 2424	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 2425	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 2426	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 2427	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15.
|- A e. V   =>   |- (A e. {B} <-> A = B)
Theorem	elsni 2428	There is only one element in a singleton.
|- (A e. {B} -> A = B)
Theorem	snidg 2429	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 2430	A class is a set iff it is a member of its singleton.
|- (A e. V <-> A e. {A})
Theorem	snid 2431	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 2432	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 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. V   =>   |- (A e. {B} <-> A = B)
Theorem	hbsn 2434	Bound-variable hypothesis builder for singletons.
|- (y e. A -> A.x y e. A)   =>   |- (y e. {A} -> A.x y e. {A})
Theorem	eltp 2435	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 2436	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 2437	Intersection with the singleton of a non-member is disjoint.
|- ((A i^i {B}) = (/) <-> -. B e. A)
Theorem	disjsn2 2438	Intersection of distinct singletons is disjoint.
|- (A =/= B -> ({A} i^i {B}) = (/))
Theorem	snprc 2439	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 2440	Special case of r19.12 1737 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 2441	Condition where a restricted class abstraction is a singleton.
|- (B e. A -> {x e. A | x = B} = {B})
Theorem	eusn 2442	Another way to express existential uniqueness of a wff: its class abstraction is a singleton.
|- (E!xph <-> E.x{x | ph} = {x})
Theorem	prcom 2443	Commutative law for unordered pairs.
|- {A, B} = {B, A}
Theorem	preq1 2444	An equality theorem for unordered pairs.
|- (A = B -> {A, C} = {B, C})
Theorem	preq2 2445	An equality theorem for unordered pairs.
|- (A = B -> {C, A} = {C, B})
Theorem	pri1gOLD 2446	One of the two elements of an unordered pair. Part of Theorem 7.6 of [Quine] p. 49.
|- (A e. C -> A e. {A, 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 e. V   &&nb