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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelon2 6201 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
(𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))

Theoremlimeq 6202 Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))

Theoremordwe 6203 Membership well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → E We 𝐴)

Theoremordtr 6204 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → Tr 𝐴)

Theoremordfr 6205 Membership is well-founded on an ordinal class. In other words, an ordinal class is well-founded. (Contributed by NM, 22-Apr-1994.)
(Ord 𝐴 → E Fr 𝐴)

Theoremordelss 6206 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
((Ord 𝐴𝐵𝐴) → 𝐵𝐴)

Theoremtrssord 6207 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Theoremordirr 6208 No ordinal class is a member of itself. In other words, the membership relation is irreflexive on ordinal classes. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994.)
(Ord 𝐴 → ¬ 𝐴𝐴)

Theoremnordeq 6209 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
((Ord 𝐴𝐵𝐴) → 𝐴𝐵)

Theoremordn2lp 6210 An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))

Theoremtz7.5 6211* A nonempty subclass of an ordinal class has a minimal element. Proposition 7.5 of [TakeutiZaring] p. 36. (Contributed by NM, 18-Feb-2004.) (Revised by David Abernethy, 16-Mar-2011.)
((Ord 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)

Theoremordelord 6212 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
((Ord 𝐴𝐵𝐴) → Ord 𝐵)

Theoremtron 6213 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
Tr On

Theoremordelon 6214 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)

Theoremonelon 6215 An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)

Theoremtz7.7 6216 A transitive class belongs to an ordinal class iff it is strictly included in it. Proposition 7.7 of [TakeutiZaring] p. 37. (Contributed by NM, 5-May-1994.)
((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴)))

Theoremordelssne 6217 For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 25-Nov-1995.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))

Theoremordelpss 6218 For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 17-Jun-1998.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))

Theoremordsseleq 6219 For ordinal classes, inclusion is equivalent to membership or equality. (Contributed by NM, 25-Nov-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Theoremordin 6220 The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)
((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))

Theoremonin 6221 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)

Theoremordtri3or 6222 A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (Contributed by NM, 10-May-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Theoremordtri1 6223 A trichotomy law for ordinals. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theoremontri1 6224 A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theoremordtri2 6225 A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))

Theoremordtri3 6226 A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by JJ, 24-Sep-2021.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))

Theoremordtri4 6227 A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))

Theoremorddisj 6228 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
(Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)

Theoremonfr 6229 The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7497 (through epweon 7496) in order to eliminate the need for the axiom of regularity. (Contributed by NM, 17-May-1994.)
E Fr On

Theoremonelpss 6230 Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))

Theoremonsseleq 6231 Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Theoremonelss 6232 An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 ∈ On → (𝐵𝐴𝐵𝐴))

Theoremordtr1 6233 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
(Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Theoremordtr2 6234 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Theoremordtr3 6235 Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.) (Proof shortened by JJ, 24-Sep-2021.)
((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))

Theoremontr1 6236 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
(𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Theoremontr2 6237 Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)
((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Theoremordunidif 6238 The union of an ordinal stays the same if a subset equal to one of its elements is removed. (Contributed by NM, 10-Dec-2004.)
((Ord 𝐴𝐵𝐴) → (𝐴𝐵) = 𝐴)

Theoremordintdif 6239 If 𝐵 is smaller than 𝐴, then it equals the intersection of the difference. Exercise 11 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)
((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴𝐵) ≠ ∅) → 𝐵 = (𝐴𝐵))

Theoremonintss 6240* 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. (Contributed by NM, 3-Oct-2003.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ On → (𝜓 {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))

Theoremoneqmini 6241* 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. (Contributed by NM, 14-Nov-2003.)
(𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))

Theoremord0 6242 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
Ord ∅

Theorem0elon 6243 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
∅ ∈ On

Theoremord0eln0 6244 A nonempty ordinal contains the empty set. (Contributed by NM, 25-Nov-1995.)
(Ord 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))

Theoremon0eln0 6245 An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.)
(𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))

Theoremdflim2 6246 An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴))

Theoreminton 6247 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
On = ∅

Theoremnlim0 6248 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
¬ Lim ∅

Theoremlimord 6249 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
(Lim 𝐴 → Ord 𝐴)

Theoremlimuni 6250 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
(Lim 𝐴𝐴 = 𝐴)

Theoremlimuni2 6251 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
(Lim 𝐴 → Lim 𝐴)

Theorem0ellim 6252 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
(Lim 𝐴 → ∅ ∈ 𝐴)

Theoremlimelon 6253 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Theoremonn0 6254 The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
On ≠ ∅

Theoremsuceq 6255 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)

Theoremelsuci 6256 Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. (Contributed by NM, 6-Jun-1994.)
(𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))

Theoremelsucg 6257 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Theoremelsuc2g 6258 Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)
(𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Theoremelsuc 6259 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Theoremelsuc2 6260 Membership in a successor. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))

Theoremnfsuc 6261 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
𝑥𝐴       𝑥 suc 𝐴

Theoremelelsuc 6262 Membership in a successor. (Contributed by NM, 20-Jun-1998.)
(𝐴𝐵𝐴 ∈ suc 𝐵)

Theoremsucel 6263* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
(suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))

Theoremsuc0 6264 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
suc ∅ = {∅}

Theoremsucprc 6265 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
𝐴 ∈ V → suc 𝐴 = 𝐴)

Theoremunisuc 6266 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. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       (Tr 𝐴 suc 𝐴 = 𝐴)

Theoremsssucid 6267 A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
𝐴 ⊆ suc 𝐴

Theoremsucidg 6268 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
(𝐴𝑉𝐴 ∈ suc 𝐴)

Theoremsucid 6269 A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
𝐴 ∈ V       𝐴 ∈ suc 𝐴

Theoremnsuceq0 6270 No successor is empty. (Contributed by NM, 3-Apr-1995.)
suc 𝐴 ≠ ∅

Theoremeqelsuc 6271 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
𝐴 ∈ V       (𝐴 = 𝐵𝐴 ∈ suc 𝐵)

Theoremiunsuc 6272* Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)

Theoremsuctr 6273 The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.)
(Tr 𝐴 → Tr suc 𝐴)

Theoremtrsuc 6274 A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)

Theoremtrsucss 6275 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
(Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))

Theoremordsssuc 6276 An ordinal is a subset of another ordinal if and only if it belongs to its successor. (Contributed by NM, 28-Nov-2003.)
((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Theoremonsssuc 6277 A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Theoremordsssuc2 6278 An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
((Ord 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Theoremonmindif 6279 When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)
((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 (𝐴 ∖ suc 𝐵))

Theoremordnbtwn 6280 There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.)
(Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

Theoremonnbtwn 6281 There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 9-Jun-1994.)
(𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

Theoremsucssel 6282 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
(𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))

Theoremorddif 6283 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
(Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))

Theoremorduniss 6284 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
(Ord 𝐴 𝐴𝐴)

Theoremordtri2or 6285 A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))

Theoremordtri2or2 6286 A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))

Theoremordtri2or3 6287 A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6286. (Contributed by David Moews, 1-May-2017.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))

Theoremordelinel 6288 The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.) (Proof shortened by JJ, 24-Sep-2021.)
((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Theoremordssun 6289 Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)
((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶)))

Theoremordequn 6290 The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵𝐴 = 𝐶)))

Theoremordun 6291 The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))

Theoremordunisssuc 6292 A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)
((𝐴 ⊆ On ∧ Ord 𝐵) → ( 𝐴𝐵𝐴 ⊆ suc 𝐵))

Theoremsuc11 6293 The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))

Theoremonordi 6294 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Ord 𝐴

Theoremontrci 6295 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Tr 𝐴

Theoremonirri 6296 An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On        ¬ 𝐴𝐴

Theoremoneli 6297 A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵 ∈ On)

Theoremonelssi 6298 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵𝐴)

Theoremonssneli 6299 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐴𝐵 → ¬ 𝐵𝐴)

Theoremonssnel2i 6300 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴 → ¬ 𝐴𝐵)

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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