Home | Intuitionistic Logic Explorer Theorem List (p. 44 of 136) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | freq2 4301 | Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) | ||
Theorem | frforeq3 4302 | Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.) |
⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇)) | ||
Theorem | nffrfor 4303 | Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝑆 ⇒ ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆 | ||
Theorem | nffr 4304 | Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 | ||
Theorem | frirrg 4305 | A well-founded relation is irreflexive. This is the case where 𝐴 exists. (Contributed by Jim Kingdon, 21-Sep-2021.) |
⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | fr0 4306 | Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.) |
⊢ 𝑅 Fr ∅ | ||
Theorem | frind 4307* | Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑)) & ⊢ (𝜒 → 𝑅 Fr 𝐴) & ⊢ (𝜒 → 𝐴 ∈ 𝑉) ⇒ ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → 𝜑) | ||
Theorem | efrirr 4308 | Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | tz7.2 4309 | Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.) |
⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)) | ||
Theorem | nfwe 4310 | Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 We 𝐴 | ||
Theorem | weeq1 4311 | Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴)) | ||
Theorem | weeq2 4312 | Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵)) | ||
Theorem | wefr 4313 | A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.) |
⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | ||
Theorem | wepo 4314 | A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.) |
⊢ ((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 Po 𝐴) | ||
Theorem | wetrep 4315* | An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.) |
⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | ||
Theorem | we0 4316 | Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.) |
⊢ 𝑅 We ∅ | ||
Syntax | word 4317 | Extend the definition of a wff to include the ordinal predicate. |
wff Ord 𝐴 | ||
Syntax | con0 4318 | Extend the definition of a class to include the class of all ordinal numbers. (The 0 in the name prevents creating a file called con.html, which causes problems in Windows.) |
class On | ||
Syntax | wlim 4319 | Extend the definition of a wff to include the limit ordinal predicate. |
wff Lim 𝐴 | ||
Syntax | csuc 4320 | Extend class notation to include the successor function. |
class suc 𝐴 | ||
Definition | df-iord 4321* | Define the ordinal predicate, which is true for a class that is transitive and whose elements are transitive. Definition of ordinal in [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". (Contributed by Jim Kingdon, 10-Oct-2018.) Use its alias dford3 4322 instead for naming consistency with set.mm. (New usage is discouraged.) |
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | ||
Theorem | dford3 4322* | Alias for df-iord 4321. Use it instead of df-iord 4321 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.) |
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | ||
Definition | df-on 4323 | Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.) |
⊢ On = {𝑥 ∣ Ord 𝑥} | ||
Definition | df-ilim 4324 | Define the limit ordinal predicate, which is true for an ordinal that has the empty set as an element and is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42, and then changes 𝐴 ≠ ∅ to ∅ ∈ 𝐴 (which would be equivalent given the law of the excluded middle, but which is not for us). (Contributed by Jim Kingdon, 11-Nov-2018.) Use its alias dflim2 4325 instead for naming consistency with set.mm. (New usage is discouraged.) |
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | ||
Theorem | dflim2 4325 | Alias for df-ilim 4324. Use it instead of df-ilim 4324 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.) |
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | ||
Definition | df-suc 4326 | Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1". Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Our definition is a generalization to classes. Although it is not conventional to use it with proper classes, it has no effect on a proper class (sucprc 4367). Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134 and the definition of successor in [Mendelson] p. 246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal"). The definition of successor of [Enderton] p. 68 denotes the operation with a plus-sign superscript. (Contributed by NM, 30-Aug-1993.) |
⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | ||
Theorem | ordeq 4327 | Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.) |
⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) | ||
Theorem | elong 4328 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴)) | ||
Theorem | elon 4329 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ On ↔ Ord 𝐴) | ||
Theorem | eloni 4330 | An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ On → Ord 𝐴) | ||
Theorem | elon2 4331 | An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.) |
⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | ||
Theorem | limeq 4332 | 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)) | ||
Theorem | ordtr 4333 | An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.) |
⊢ (Ord 𝐴 → Tr 𝐴) | ||
Theorem | ordelss 4334 | An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) | ||
Theorem | trssord 4335 | A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.) |
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴) | ||
Theorem | ordelord 4336 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) | ||
Theorem | tron 4337 | The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.) |
⊢ Tr On | ||
Theorem | ordelon 4338 | An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | ||
Theorem | onelon 4339 | 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 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | ||
Theorem | ordin 4340 | 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 (𝐴 ∩ 𝐵)) | ||
Theorem | onin 4341 | The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On) | ||
Theorem | onelss 4342 | 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 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | ||
Theorem | ordtr1 4343 | Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) |
⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | ||
Theorem | ontr1 4344 | Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.) |
⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | ||
Theorem | onintss 4345* | 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ On → (𝜓 → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴)) | ||
Theorem | ord0 4346 | The empty set is an ordinal class. (Contributed by NM, 11-May-1994.) |
⊢ Ord ∅ | ||
Theorem | 0elon 4347 | The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.) |
⊢ ∅ ∈ On | ||
Theorem | inton 4348 | The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.) |
⊢ ∩ On = ∅ | ||
Theorem | nlim0 4349 | The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ ¬ Lim ∅ | ||
Theorem | limord 4350 | A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.) |
⊢ (Lim 𝐴 → Ord 𝐴) | ||
Theorem | limuni 4351 | A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.) |
⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | ||
Theorem | limuni2 4352 | The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.) |
⊢ (Lim 𝐴 → Lim ∪ 𝐴) | ||
Theorem | 0ellim 4353 | A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.) |
⊢ (Lim 𝐴 → ∅ ∈ 𝐴) | ||
Theorem | limelon 4354 | A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.) |
⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On) | ||
Theorem | onn0 4355 | The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.) |
⊢ On ≠ ∅ | ||
Theorem | onm 4356 | The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.) |
⊢ ∃𝑥 𝑥 ∈ On | ||
Theorem | suceq 4357 | Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) | ||
Theorem | elsuci 4358 | 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 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | elsucg 4359 | Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | elsuc2g 4360 | Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | elsuc 4361 | Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | elsuc2 4362 | Membership in a successor. (Contributed by NM, 15-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | ||
Theorem | nfsuc 4363 | Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 suc 𝐴 | ||
Theorem | elelsuc 4364 | Membership in a successor. (Contributed by NM, 20-Jun-1998.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵) | ||
Theorem | sucel 4365* | Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.) |
⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))) | ||
Theorem | suc0 4366 | The successor of the empty set. (Contributed by NM, 1-Feb-2005.) |
⊢ suc ∅ = {∅} | ||
Theorem | sucprc 4367 | A proper class is its own successor. (Contributed by NM, 3-Apr-1995.) |
⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | ||
Theorem | unisuc 4368 | 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) | ||
Theorem | unisucg 4369 | 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 Jim Kingdon, 18-Aug-2019.) |
⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)) | ||
Theorem | sssucid 4370 | 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 | ||
Theorem | sucidg 4371 | Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | ||
Theorem | sucid 4372 | 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 | ||
Theorem | nsuceq0g 4373 | No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.) |
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ≠ ∅) | ||
Theorem | eqelsuc 4374 | A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵) | ||
Theorem | iunsuc 4375* | 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 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶) | ||
Theorem | suctr 4376 | The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) |
⊢ (Tr 𝐴 → Tr suc 𝐴) | ||
Theorem | trsuc 4377 | 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 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | ||
Theorem | trsucss 4378 | A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.) |
⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴)) | ||
Theorem | sucssel 4379 | A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.) |
⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) | ||
Theorem | orduniss 4380 | An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.) |
⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴) | ||
Theorem | onordi 4381 | An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ Ord 𝐴 | ||
Theorem | ontrci 4382 | An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ Tr 𝐴 | ||
Theorem | oneli 4383 | 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 ⇒ ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On) | ||
Theorem | onelssi 4384 | A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴) | ||
Theorem | onelini 4385 | An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴)) | ||
Theorem | oneluni 4386 | An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴) | ||
Theorem | onunisuci 4387 | An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ ∪ suc 𝐴 = 𝐴 | ||
Axiom | ax-un 4388* |
Axiom of Union. An axiom of Intuitionistic Zermelo-Fraenkel set theory.
It states that a set 𝑦 exists that includes the union of a
given set
𝑥 i.e. the collection of all members of
the members of 𝑥. The
variant axun2 4390 states that the union itself exists. A
version with the
standard abbreviation for union is uniex2 4391. A version using class
notation is uniex 4392.
This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4081), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 264). The union of a class df-uni 3769 should not be confused with the union of two classes df-un 3102. Their relationship is shown in unipr 3782. (Contributed by NM, 23-Dec-1993.) |
⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
Theorem | zfun 4389* | Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) |
⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
Theorem | axun2 4390* | A variant of the Axiom of Union ax-un 4388. For any set 𝑥, there exists a set 𝑦 whose members are exactly the members of the members of 𝑥 i.e. the union of 𝑥. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) | ||
Theorem | uniex2 4391* | The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃𝑦 𝑦 = ∪ 𝑥 | ||
Theorem | uniex 4392 | The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2715), then the union of 𝐴 is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∪ 𝐴 ∈ V | ||
Theorem | vuniex 4393 | The union of a setvar is a set. (Contributed by BJ, 3-May-2021.) |
⊢ ∪ 𝑥 ∈ V | ||
Theorem | uniexg 4394 | The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to make the theorem more general and thus shorten some proofs; obviously the universal class constant V is one possible substitution for class variable 𝑉. (Contributed by NM, 25-Nov-1994.) |
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | ||
Theorem | unex 4395 | The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∪ 𝐵) ∈ V | ||
Theorem | unexb 4396 | Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) | ||
Theorem | unexg 4397 | A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | ||
Theorem | tpexg 4398 | An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.) |
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {𝐴, 𝐵, 𝐶} ∈ V) | ||
Theorem | unisn3 4399* | Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.) |
⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴) | ||
Theorem | abnexg 4400* | Sufficient condition for a class abstraction to be a proper class. The class 𝐹 can be thought of as an expression in 𝑥 and the abstraction appearing in the statement as the class of values 𝐹 as 𝑥 varies through 𝐴. Assuming the antecedents, if that class is a set, then so is the "domain" 𝐴. The converse holds without antecedent, see abrexexg 6056. Note that the second antecedent ∀𝑥 ∈ 𝐴𝑥 ∈ 𝐹 cannot be translated to 𝐴 ⊆ 𝐹 since 𝐹 may depend on 𝑥. In applications, one may take 𝐹 = {𝑥} or 𝐹 = 𝒫 𝑥 (see snnex 4402 and pwnex 4403 respectively, proved from abnex 4401, which is a consequence of abnexg 4400 with 𝐴 = V). (Contributed by BJ, 2-Dec-2021.) |
⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → 𝐴 ∈ V)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |