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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | uniex2 4201* | 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 4202 | The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2578), 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 | uniexg 4203 | 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 4204 | 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 4205 | 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 4206 | 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 4207 | An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.) |
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {𝐴, 𝐵, 𝐶} ∈ V) | ||
Theorem | unisn3 4208* | 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 | snnex 4209* | The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) |
⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | ||
Theorem | opeluu 4210 | Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶)) | ||
Theorem | uniuni 4211* | Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.) |
⊢ ∪ ∪ 𝐴 = ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)} | ||
Theorem | eusv1 4212* | Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.) |
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴) | ||
Theorem | eusvnf 4213* | Even if 𝑥 is free in 𝐴, it is effectively bound when 𝐴(𝑥) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴) | ||
Theorem | eusvnfb 4214* | Two ways to say that 𝐴(𝑥) is a set expression that does not depend on 𝑥. (Contributed by Mario Carneiro, 18-Nov-2016.) |
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V)) | ||
Theorem | eusv2i 4215* | Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.) |
⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴) | ||
Theorem | eusv2nf 4216* | Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by Mario Carneiro, 18-Nov-2016.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴) | ||
Theorem | eusv2 4217* | Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴) | ||
Theorem | reusv1 4218* | Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) | ||
Theorem | reusv3i 4219* | Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.) |
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷) ⇒ ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) | ||
Theorem | reusv3 4220* | Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 4218 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.) |
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷) ⇒ ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) | ||
Theorem | alxfr 4221* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
Theorem | ralxfrd 4222* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | rexxfrd 4223* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | ralxfr2d 4224* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | rexxfr2d 4225* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | ralxfr 4226* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓) | ||
Theorem | ralxfrALT 4227* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. This proof does not use ralxfrd 4222. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓) | ||
Theorem | rexxfr 4228* | Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐶 𝜓) | ||
Theorem | rabxfrd 4229* | 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 4230* | 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 4231* | 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 4232* | 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 4233 | 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 4234 | 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 4235 | 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 4236 | 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 4237 | 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 4238 | 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 4239* | 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 4240 | 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 4241 | 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 4242 | 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 4243 | 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 4244 | 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 4245 | 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 4246 | 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 4247 | An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | ||
Theorem | onuni 4248 | The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) | ||
Theorem | orduni 4249 | The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.) |
⊢ (Ord 𝐴 → Ord ∪ 𝐴) | ||
Theorem | bm2.5ii 4250* | 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 4251 | A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | ||
Theorem | sucexg 4252 | The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) | ||
Theorem | sucex 4253 | The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ suc 𝐴 ∈ V | ||
Theorem | ordsucim 4254 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
⊢ (Ord 𝐴 → Ord suc 𝐴) | ||
Theorem | suceloni 4255 | 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 4256 | The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴)) | ||
Theorem | sucelon 4257 | The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) | ||
Theorem | ordsucss 4258 | 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 4259 | 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 4260 | 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 4261* | 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 4262 | Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4283. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6101. (Contributed by Jim Kingdon, 17-Jul-2019.) |
⊢ (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵)) | ||
Theorem | onsucsssucr 4263 | The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4280. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵)) | ||
Theorem | sucunielr 4264 | Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4284. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵) | ||
Theorem | unon 4265 | 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 4266* | 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 4267 | The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
⊢ Lim On | ||
Theorem | ordunisuc2r 4268* | 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 4269 | An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.) |
⊢ 𝐴 ∈ On ⇒ ⊢ 𝐴 ⊆ On | ||
Theorem | onsuci 4270 | 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 4271* | 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 4272 | 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 4273 | Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4275 or weak linearity in ordsoexmid 4314) 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 4274* | Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4275 or weak linearity in ordsoexmid 4314) 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 4275* |
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 4276* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑦 ⊆ 𝑥) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | 2ordpr 4277 | Version of 2on 6040 with the definition of 2_{𝑜} expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.) |
⊢ Ord {∅, {∅}} | ||
Theorem | ontr2exmid 4278* | 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 4279* | 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 4280* | The converse of onsucsssucr 4263 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 4281* | Lemma for onsucelsucexmid 4283. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} | ||
Theorem | onsucelsucexmidlem 4282* | Lemma for onsucelsucexmid 4283. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5531), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4273. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On | ||
Theorem | onsucelsucexmid 4283* | The converse of onsucelsucr 4262 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 4262 does hold, as seen at nnsucelsuc 6101. (Contributed by Jim Kingdon, 2-Aug-2019.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | ordsucunielexmid 4284* | The converse of sucunielr 4264 (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 4285* | Lemma for regexmid 4288. 𝐴 is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} ⇒ ⊢ ∃𝑦 𝑦 ∈ 𝐴 | ||
Theorem | regexmidlem1 4286* | Lemma for regexmid 4288. 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 4287* | Lemma for reg2exmid 4289. 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 4288* |
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 4290. (Contributed by Jim Kingdon, 3-Sep-2019.) |
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | reg2exmid 4289* | 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 4290* |
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 4291* | ∈-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 4292* | 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 4293 | 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 4294 | 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 4295 | 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 4296 | 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 4297 | 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 4298 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | ||
Theorem | orddif 4299 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) | ||
Theorem | elirrv 4300 | 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 |
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