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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rabxfrd 4201* | Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜒. (Contributed by NM, 16-Jan-2012.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑦𝐶 & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})) | ||
Theorem | rabxfr 4202* | Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜑. (Contributed by NM, 10-Jun-2005.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑦𝐶 & ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) ⇒ ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) | ||
Theorem | reuhypd 4203* | A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | ||
Theorem | reuhyp 4204* | A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶) & ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) ⇒ ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | ||
Theorem | uniexb 4205 | The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.) |
⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | ||
Theorem | pwexb 4206 | The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.) |
⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | ||
Theorem | univ 4207 | The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) |
⊢ ∪ V = V | ||
Theorem | eldifpw 4208 | Membership in a power class difference. (Contributed by NM, 25-Mar-2007.) |
⊢ 𝐶 ∈ V ⇒ ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵)) | ||
Theorem | op1stb 4209 | Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴 | ||
Theorem | op1stbg 4210 | Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴) | ||
Theorem | iunpw 4211* | An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 ↔ 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥) | ||
Theorem | ordon 4212 | The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.) |
⊢ Ord On | ||
Theorem | ssorduni 4213 | The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | ||
Theorem | ssonuni 4214 | The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) | ||
Theorem | ssonunii 4215 | The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On) | ||
Theorem | onun2 4216 | The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) | ||
Theorem | onun2i 4217 | The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.) |
⊢ 𝐴 ∈ On & ⊢ 𝐵 ∈ On ⇒ ⊢ (𝐴 ∪ 𝐵) ∈ On | ||
Theorem | ordsson 4218 | Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) |
⊢ (Ord 𝐴 → 𝐴 ⊆ On) | ||
Theorem | onss 4219 | An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | ||
Theorem | onuni 4220 | The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) | ||
Theorem | orduni 4221 | The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.) |
⊢ (Ord 𝐴 → Ord ∪ 𝐴) | ||
Theorem | bm2.5ii 4222* | Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) | ||
Theorem | sucexb 4223 | A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | ||
Theorem | sucexg 4224 | The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) | ||
Theorem | sucex 4225 | The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ suc 𝐴 ∈ V | ||
Theorem | ordsucim 4226 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
⊢ (Ord 𝐴 → Ord suc 𝐴) | ||
Theorem | suceloni 4227 | The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | ||
Theorem | ordsucg 4228 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴)) | ||
Theorem | sucelon 4229 | The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) | ||
Theorem | ordsucss 4230 | The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | ||
Theorem | ordelsuc 4231 | A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.) |
⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | ||
Theorem | onsucssi 4232 | A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.) |
⊢ 𝐴 ∈ On & ⊢ 𝐵 ∈ On ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵) | ||
Theorem | onsucmin 4233* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥}) | ||
Theorem | onsucelsucr 4234 | Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4255. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6070. (Contributed by Jim Kingdon, 17-Jul-2019.) |
⊢ (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵)) | ||
Theorem | onsucsssucr 4235 | The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4252. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵)) | ||
Theorem | sucunielr 4236 | Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4256. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵) | ||
Theorem | unon 4237 | The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) |
⊢ ∪ On = On | ||
Theorem | onuniss2 4238* | The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ (𝐴 ∈ On → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴) | ||
Theorem | limon 4239 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
⊢ Lim On | ||
Theorem | ordunisuc2r 4240* | An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.) |
⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 = ∪ 𝐴)) | ||
Theorem | onssi 4241 | An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ 𝐴 ⊆ On | ||
Theorem | onsuci 4242 | The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ suc 𝐴 ∈ On | ||
Theorem | onintonm 4243* | The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.) |
⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ On) | ||
Theorem | onintrab2im 4244 | An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | ||
Theorem | ordtriexmidlem 4245 | Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4247 or weak linearity in ordsoexmid 4286) with a proposition 𝜑. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.) |
⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On | ||
Theorem | ordtriexmidlem2 4246* | Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4247 or weak linearity in ordsoexmid 4286) with a proposition 𝜑. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.) |
⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑) | ||
Theorem | ordtriexmid 4247* |
Ordinal trichotomy implies the law of the excluded middle (that is,
decidability of an arbitrary proposition).
This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | ordtri2orexmid 4248* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | 2ordpr 4249 | Version of 2on 6009 with the definition of 2_{𝑜} expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.) |
⊢ Ord {∅, {∅}} | ||
Theorem | ontr2exmid 4250* | An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.) |
⊢ ∀𝑥 ∈ On ∀𝑦∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | ordtri2or2exmidlem 4251* | A set which is 2_{𝑜} if 𝜑 or ∅ if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.) |
⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On | ||
Theorem | onsucsssucexmid 4252* | The converse of onsucsssucr 4235 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 → suc 𝑥 ⊆ suc 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | onsucelsucexmidlem1 4253* | Lemma for onsucelsucexmid 4255. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} | ||
Theorem | onsucelsucexmidlem 4254* | Lemma for onsucelsucexmid 4255. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5503), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4245. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On | ||
Theorem | onsucelsucexmid 4255* | The converse of onsucelsucr 4234 implies excluded middle. On the other hand, if 𝑦 is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4234 does hold, as seen at nnsucelsuc 6070. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | ordsucunielexmid 4256* | The converse of sucunielr 4236 (where 𝐵 is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | regexmidlemm 4257* | Lemma for regexmid 4260. 𝐴 is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} ⇒ ⊢ ∃𝑦 𝑦 ∈ 𝐴 | ||
Theorem | regexmidlem1 4258* | Lemma for regexmid 4260. If 𝐴 has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} ⇒ ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)) → (𝜑 ∨ ¬ 𝜑)) | ||
Theorem | reg2exmidlema 4259* | Lemma for reg2exmid 4261. If 𝐴 has a minimal element (expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} ⇒ ⊢ (∃𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 → (𝜑 ∨ ¬ 𝜑)) | ||
Theorem | regexmid 4260* |
The axiom of foundation implies excluded middle.
By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by ∈). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4262. (Contributed by Jim Kingdon, 3-Sep-2019.) |
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | reg2exmid 4261* | If any inhabited set has a minimal element (when expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
⊢ ∀𝑧(∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Axiom | ax-setind 4262* |
Axiom of ∈-Induction (also known as set
induction). An axiom of
Intuitionistic Zermelo-Fraenkel set theory. Axiom 9 of [Crosilla] p.
"Axioms of CZF and IZF". This replaces the Axiom of
Foundation (also
called Regularity) from Zermelo-Fraenkel set theory.
For more on axioms which might be adopted which are incompatible with this axiom (that is, Non-wellfounded Set Theory but in the absence of excluded middle), see Chapter 20 of [AczelRathjen], p. 183. (Contributed by Jim Kingdon, 19-Oct-2018.) |
⊢ (∀𝑎(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑) | ||
Theorem | setindel 4263* | ∈-Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.) |
⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝑆 = V) | ||
Theorem | setind 4264* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V) | ||
Theorem | setind2 4265 | Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.) |
⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V) | ||
Theorem | elirr 4266 | No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) |
⊢ ¬ 𝐴 ∈ 𝐴 | ||
Theorem | ordirr 4267 | Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. (Contributed by NM, 2-Jan-1994.) |
⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | nordeq 4268 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) | ||
Theorem | ordn2lp 4269 | 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 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | ||
Theorem | orddisj 4270 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | ||
Theorem | orddif 4271 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) | ||
Theorem | elirrv 4272 | The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.) |
⊢ ¬ 𝑥 ∈ 𝑥 | ||
Theorem | sucprcreg 4273 | A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.) |
⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴) | ||
Theorem | ruv 4274 | The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | ||
Theorem | ruALT 4275 | Alternate proof of Russell's Paradox ru 2763, simplified using (indirectly) the Axiom of Set Induction ax-setind 4262. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Theorem | onprc 4276 | No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4212), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.) |
⊢ ¬ On ∈ V | ||
Theorem | sucon 4277 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
⊢ suc On = On | ||
Theorem | en2lp 4278 | No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 27-Nov-2018.) |
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) | ||
Theorem | preleq 4279 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | opthreg 4280 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4262 (via the preleq 4279 step). See df-op 3384 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | suc11g 4281 | The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | suc11 4282 | 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 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | dtruex 4283* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3942 can also be summarized as "at least two sets exist", the difference is that dtruarb 3942 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific 𝑦, we can construct a set 𝑥 which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.) |
⊢ ∃𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | dtru 4284* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4283. (Contributed by Jim Kingdon, 29-Dec-2018.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | eunex 4285 | Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) | ||
Theorem | ordsoexmid 4286 | Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.) |
⊢ E Or On ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | ordsuc 4287 | The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.) |
⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | ||
Theorem | onsucuni2 4288 | A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴) | ||
Theorem | 0elsucexmid 4289* | If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥 ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | nlimsucg 4290 | A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴) | ||
Theorem | ordpwsucss 4291 |
The collection of ordinals in the power class of an ordinal is a
superset of its successor.
We can think of (𝒫 𝐴 ∩ On) as another possible definition of successor, which would be equivalent to df-suc 4108 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4169) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4238). Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4294). (Contributed by Jim Kingdon, 21-Jul-2019.) |
⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On)) | ||
Theorem | onnmin 4292 | No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.) |
⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴) | ||
Theorem | ssnel 4293 | Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.) |
⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴) | ||
Theorem | ordpwsucexmid 4294* | The subset in ordpwsucss 4291 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
⊢ ∀𝑥 ∈ On suc 𝑥 = (𝒫 𝑥 ∩ On) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | onpsssuc 4295 | An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴) | ||
Theorem | ordtri2or2exmid 4296* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | onintexmid 4297* | If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.) |
⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | zfregfr 4298 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
⊢ E Fr 𝐴 | ||
Theorem | ordfr 4299 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
⊢ (Ord 𝐴 → E Fr 𝐴) | ||
Theorem | ordwe 4300 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
⊢ (Ord 𝐴 → E We 𝐴) |
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