mmtheorems31 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3001-3100 - Page 31 of 107

Type	Label	Description
Statement
Theorem	ordunidif 3001	The union of an ordinal stays the same if a subset equal to one of its elements is removed.
|- ((Ord A /\ B e. A) -> U.(A \ B) = U.A)
Theorem	onint 3002	The intersection (infimum) of a non-empty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45.
|- ((A (_ On /\ A =/= (/)) -> |^|A e. A)
Theorem	onint0 3003	The intersection of a class of ordinal numbers is zero iff the class contains zero.
|- (A (_ On -> (|^|A = (/) <-> (/) e. A))
Theorem	onssmin 3004	A non-empty class of ordinal numbers has a smallest member. Exercise 9 of [TakeutiZaring] p. 40.
|- ((A (_ On /\ A =/= (/)) -> E.x e. A A.y e. A x (_ y)
Theorem	onminsb 3005	If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228.
|- (ps -> A.xps)   &   |- (x = |^|{x e. On | ph} -> (ph <-> ps))   =>   |- (E.x e. On ph -> ps)
Theorem	onminesb 3006	If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses explicit substitution. Theorem Schema 62 of [Suppes] p. 228.
|- (E.x e. On ph -> [|^|{x e. On | ph} / x]ph)
Theorem	onintss 3007	If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228.
|- (x = A -> (ph <-> ps))   =>   |- (A e. On -> (ps -> |^|{x e. On | ph} (_ A))
Theorem	oninton 3008	The intersection of a non-empty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44.
|- ((A (_ On /\ A =/= (/)) -> |^|A e. On)
Theorem	onintrab 3009	The intersection of a class of ordinal numbers exists iff it is an ordinal number.
|- (|^|{x e. On | ph} e. V <-> |^|{x e. On | ph} e. On)
Theorem	onintrab2 3010	An existence condition equivalent to an intersection's being an ordinal number.
|- (E.x e. On ph <-> |^|{x e. On | ph} e. On)
Theorem	onnmin 3011	No member of a set of ordinal numbers belongs to its minimum.
|- ((A (_ On /\ B e. A) -> -. B e. |^|A)
Theorem	onnminsb 3012	An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. ps is the wff resulting from the substitution of A for x in wff ph.
|- (x = A -> (ph <-> ps))   =>   |- (A e. On -> (A e. |^|{x e. On | ph} -> -. ps))
Theorem	oneqmini 3013	A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection.
|- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))
Theorem	oneqmin 3014	A way to show that an ordinal number equals the minimum of a non-empty collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection.
|- ((B (_ On /\ B =/= (/)) -> (A = |^|B <-> (A e. B /\ A.x e. A -. x e. B)))
Theorem	bm2.5ii 3015	Problem 2.5(ii) of [BellMachover] p. 471.
|- A e. V   =>   |- (A (_ On -> U.A = |^|{x e. On | A.y e. A y (_ x})
Theorem	onminex 3016	If a wff is true for an ordinal number, there is a smallest ordinal number for which it is true.
|- (x = y -> (ph <-> ps))   =>   |- (E.x e. On ph -> E.x e. On (ph /\ A.y e. x -. ps))
Theorem	ord0 3017	The empty set is an ordinal class.
|- Ord (/)
Theorem	0elon 3018	The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193.
|- (/) e. On
Theorem	ord0eln0 3019	A non-empty ordinal contains the empty set.
|- (Ord A -> ((/) e. A <-> A =/= (/)))
Theorem	on0eln0 3020	An ordinal number contains zero iff it is nonzero.
|- (A e. On -> ((/) e. A <-> A =/= (/)))
Theorem	dflim2 3021	An alternate definition of a limit ordinal.
|- (Lim A <-> (Ord A /\ (/) e. A /\ A = U.A))
Theorem	inton 3022	The intersection of the class of ordinal numbers is the empty set.
|- |^|On = (/)
Theorem	nlim0 3023	The empty set is not a limit ordinal.
|- -. Lim (/)
Theorem	limord 3024	A limit ordinal is ordinal.
|- (Lim A -> Ord A)
Theorem	limuni 3025	A limit ordinal is its own supremum (union).
|- (Lim A -> A = U.A)
Theorem	limuni2 3026	The union of a limit ordinal is a limit ordinal.
|- (Lim A -> Lim U.A)
Theorem	0ellim 3027	A limit ordinal contains the empty set.
|- (Lim A -> (/) e. A)
Theorem	limelon 3028	A limit ordinal class that is also a set is an ordinal number.
|- ((A e. B /\ Lim A) -> A e. On)
Theorem	onne0 3029	The class of all ordinal numbers in not empty.
|- On =/= (/)
Theorem	suceq 3030	Equality of successors.
|- (A = B -> suc A = suc B)
Theorem	elsuci 3031	Membership in a successor. This one-way implication does not require that either A or B be sets.
|- (A e. suc B -> (A e. B \/ A = B))
Theorem	elsucg 3032	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17.
|- (A e. C -> (A e. suc B <-> (A e. B \/ A = B)))
Theorem	elsuc2g 3033	Variant of membership in a successor, requiring that B rather than A be a set.
|- (B e. C -> (A e. suc B <-> (A e. B \/ A = B)))
Theorem	elsuc 3034	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17.
|- A e. V   =>   |- (A e. suc B <-> (A e. B \/ A = B))
Theorem	elsuc2 3035	Membership in a successor.
|- A e. V   =>   |- (B e. suc A <-> (B e. A \/ B = A))
Theorem	hbsuc 3036	Bound-variable hypothesis builder for successor.
|- (y e. A -> A.x y e. A)   =>   |- (y e. suc A -> A.x y e. suc A)
Theorem	elelsuc 3037	Membership in a successor.
|- (A e. B -> A e. suc B)
Theorem	sucel 3038	Membership of a successor in another class.
|- (suc A e. B <-> E.x e. B A.y(y e. x <-> (y e. A \/ y = A)))
Theorem	suc0 3039	The successor of the empty set.
|- suc (/) = {(/)}
Theorem	sucprc 3040	A proper class is its own successor.
|- (-. A e. V -> suc A = A)
Theorem	sucon 3041	The class of all ordinal numbers is its own successor.
|- suc On = On
Theorem	unisuc 3042	A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73.
|- A e. V   =>   |- (Tr A <-> U.suc A = A)
Theorem	sssucid 3043	A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes).
|- A (_ suc A
Theorem	sucexb 3044	A successor exists iff its class argument exists.
|- (A e. V <-> suc A e. V)
Theorem	sucexg 3045	The successor of a set is a set (generalization).
|- (A e. B -> suc A e. V)
Theorem	sucex 3046	The successor of a set is a set.
|- A e. V   =>   |- suc A e. V
Theorem	sucid 3047	A set belongs to its successor.
|- A e. V   =>   |- A e. suc A
Theorem	sucidg 3048	Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized).
|- (A e. B -> A e. suc A)
Theorem	nsuceq0 3049	No successor is empty.
|- suc A =/= (/)
Theorem	eqelsuc 3050	A set belongs to the successor of an equal set.
|- A e. V   =>   |- (A = B -> A e. suc B)
Theorem	trsuc 3051	A set whose successor belongs to a transitive class also belongs.
|- ((Tr A /\ suc B e. A) -> B e. A)
Theorem	trsucss 3052	A member of the successor of a transitive class is a subclass of it.
|- (Tr A -> (B e. suc A -> B (_ A))
Theorem	ordsssuc 3053	A subset of an ordinal belongs to its successor.
|- ((A e. On /\ Ord B) -> (A (_ B <-> A e. suc B))
Theorem	onsssuc 3054	A subset of an ordinal number belongs to its successor.
|- ((A e. On /\ B e. On) -> (A (_ B <-> A e. suc B))
Theorem	ordsssuc2 3055	An ordinal subset of an ordinal number belongs to its successor.
|- ((Ord A /\ B e. On) -> (A (_ B <-> A e. suc B))
Theorem	onmindif 3056	When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass.
|- ((A (_ On /\ B e. On) -> B e. |^|(A \ suc B))
Theorem	onmindif2 3057	The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed.
|- ((A (_ On /\ A =/= (/)) -> |^|A e. |^|(A \ {|^|A}))
Theorem	suceloni 3058	The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41.
|- (A e. On -> suc A e. On)
Theorem	ordnbtwn 3059	There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41.
|- (Ord A -> -. (A e. B /\ B e. suc A))
Theorem	onnbtwn 3060	There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41.
|- (A e. On -> -. (A e. B /\ B e. suc A))
Theorem	ordsuc 3061	The successor of an ordinal class is ordinal.
|- (Ord A <-> Ord suc A)
Theorem	ordpwsuc 3062	The collection of ordinals in the power class of an ordinal is its successor.
|- (Ord A -> (P~A i^i On) = suc A)
Theorem	onpwsuc 3063	The collection of ordinal numbers in the power set of an ordinal number is its successor.
|- (A e. On -> (P~A i^i On) = suc A)
Theorem	sucelon 3064	The successor of an ordinal number is an ordinal number.
|- (A e. On <-> suc A e. On)
Theorem	ordsucss 3065	The successor of an element of an ordinal class is a subset of it.
|- (Ord B -> (A e. B -> suc A (_ B))
Theorem	sucssel 3066	A set whose successor is a subset of another class is a member of that class.
|- (A e. C -> (suc A (_ B -> A e. B))
Theorem	ordelsuc 3067	A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse.
|- ((A e. C /\ Ord B) -> (A e. B <-> suc A (_ B))
Theorem	onsucmin 3068	The successor of an ordinal number is the smallest larger ordinal number.
|- (A e. On -> suc A = |^|{x e. On | A e. x})
Theorem	ordsucelsuc 3069	Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42.
|- (Ord B -> (A e. B <-> suc A e. suc B))
Theorem	ordsucsssuc 3070	The subclass relationship between two ordinal classes is inherited by their successors.
|- ((Ord A /\ Ord B) -> (A (_ B <-> suc A (_ suc B))
Theorem	orddif 3071	Ordinal derived from its successor.
|- (Ord A -> A = (suc A \ {A}))
Theorem	orduniss 3072	An ordinal class includes its union.
|- (Ord A -> U.A (_ A)
Theorem	ordtri2or 3073	A trichotomy law for ordinal classes.
|- ((Ord A /\ Ord B) -> (A e. B \/ B (_ A))
Theorem	ordtri2or2 3074	A trichotomy law for ordinal classes.
|- ((Ord A /\ Ord B) -> (A (_ B \/ B (_ A))
Theorem	ordssun 3075	Property of a subclass of the maximum (i.e. union) of two ordinals.
|- ((Ord B /\ Ord C) -> (A (_ (B u. C) <-> (A (_ B \/ A (_ C)))
Theorem	ordequn 3076	The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40.
|- ((Ord B /\ Ord C) -> (A = (B u. C) -> (A = B \/ A = C)))
Theorem	ordun 3077	The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40.
|- ((Ord A /\ Ord B) -> Ord (A u. B))
Theorem	ordsucun 3078	The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors.
|- ((Ord A /\ Ord B) -> suc (A u. B) = (suc A u. suc B))
Theorem	ordunisssuc 3079	A subclass relationship for union and successor of ordinal classes.
|- ((A (_ On /\ Ord B) -> (U.A (_ B <-> A (_ suc B))
Theorem	ordunel 3080	The maximum of two ordinals belongs to a third if each of them do.
|- ((Ord A /\ B