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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cbvopab 5101* | Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.) |
⊢ Ⅎ𝑧𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} | ||
Theorem | cbvopabv 5102* | Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} | ||
Theorem | cbvopab1 5103* | Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2379. See cbvopab1g 5104 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.) |
⊢ Ⅎ𝑧𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜓} | ||
Theorem | cbvopab1g 5104* | Change first bound variable in an ordered-pair class abstraction, using explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. See cbvopab1 5103 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑧𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜓} | ||
Theorem | cbvopab2 5105* | Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.) |
⊢ Ⅎ𝑧𝜑 & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑧〉 ∣ 𝜓} | ||
Theorem | cbvopab1s 5106* | Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.) |
⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} | ||
Theorem | cbvopab1v 5107* | Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜓} | ||
Theorem | cbvopab2v 5108* | Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.) |
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑧〉 ∣ 𝜓} | ||
Theorem | unopab 5109 | Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ∪ {〈𝑥, 𝑦〉 ∣ 𝜓}) = {〈𝑥, 𝑦〉 ∣ (𝜑 ∨ 𝜓)} | ||
Syntax | cmpt 5110 | Extend the definition of a class to include maps-to notation for defining a function via a rule. |
class (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Definition | df-mpt 5111* | Define maps-to notation for defining a function via a rule. Read as "the function which maps 𝑥 (in 𝐴) to 𝐵(𝑥)". The class expression 𝐵 is the value of the function at 𝑥 and normally contains the variable 𝑥. An example is the square function for complex numbers, (𝑥 ∈ ℂ ↦ (𝑥↑2)). Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.) |
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | ||
Theorem | mpteq12df 5112 | An equality inference for the maps-to notation. Compare mpteq12dv 5115. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | mpteq12f 5113 | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | mpteq12dva 5114* | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | mpteq12dv 5115* | An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) Drop ax-10 2142 while shortening its proof. (Revised by Steven Nguyen and Gino Giotto, 1-Dec-2023.) |
⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | mpteq12dvOLD 5116* | Obsolete version of mpteq12dv 5115 as of 1-Dec-2023. (Contributed by NM, 24-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | mpteq12 5117* | An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.) |
⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | mpteq1 5118* | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | mpteq1d 5119* | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
Theorem | mpteq1i 5120* | An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | ||
Theorem | mpteq2ia 5121 | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | mpteq2i 5122 | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | mpteq12i 5123 | An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.) |
⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷) | ||
Theorem | mpteq2da 5124 | Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | ||
Theorem | mpteq2dva 5125* | Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | ||
Theorem | mpteq2dv 5126* | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | ||
Theorem | nfmpt 5127* | Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | nfmpt1 5128 | Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.) |
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | cbvmptf 5129* | Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) Add disjoint variable condition to avoid ax-13 2379. See cbvmptfg 5130 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptfg 5130 | Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2379. See cbvmptf 5129 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmpt 5131* | Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) Add disjoint variable condition to avoid ax-13 2379. See cbvmptg 5132 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptg 5132* | Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2379. See cbvmpt 5131 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by NM, 11-Sep-2011.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptv 5133* | Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid ax-13 2379. See cbvmptvg 5134 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptvg 5134* | Rule to change the bound variable in a maps-to function, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. See cbvmptv 5133 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by Mario Carneiro, 19-Feb-2013.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | mptv 5135* | Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} | ||
Syntax | wtr 5136 | Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35. |
wff Tr 𝐴 | ||
Definition | df-tr 5137 | Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5939). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 5138 (which is suggestive of the word "transitive"), dftr3 5140, dftr4 5141, dftr5 5139, and (when 𝐴 is a set) unisuc 6235. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.) |
⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | ||
Theorem | dftr2 5138* | An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.) |
⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) | ||
Theorem | dftr5 5139* | An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.) |
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | ||
Theorem | dftr3 5140* | An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.) |
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) | ||
Theorem | dftr4 5141 | An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.) |
⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴) | ||
Theorem | treq 5142 | Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.) |
⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵)) | ||
Theorem | trel 5143 | In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | ||
Theorem | trel3 5144 | In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) |
⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | ||
Theorem | trss 5145 | An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) (Proof shortened by JJ, 26-Jul-2021.) |
⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | ||
Theorem | trin 5146 | The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.) |
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) | ||
Theorem | tr0 5147 | The empty set is transitive. (Contributed by NM, 16-Sep-1993.) |
⊢ Tr ∅ | ||
Theorem | trv 5148 | The universe is transitive. (Contributed by NM, 14-Sep-2003.) |
⊢ Tr V | ||
Theorem | triun 5149 | An indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | truni 5150* | The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.) |
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴) | ||
Theorem | triin 5151 | An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.) |
⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | trint 5152* | The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by BJ, 3-Oct-2022.) |
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴) | ||
Theorem | trintss 5153 | Any nonempty transitive class includes its intersection. Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011.) (Proof shortened by Andrew Salmon, 14-Nov-2011.) |
⊢ ((Tr 𝐴 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝐴) | ||
Axiom | ax-rep 5154* |
Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory.
Axiom 5 of [TakeutiZaring] p. 19.
It tells us that the image of any set
under a function is also a set (see the variant funimaex 6411). Although
𝜑 may be any wff whatsoever, this
axiom is useful (i.e. its
antecedent is satisfied) when we are given some function and 𝜑
encodes the predicate "the value of the function at 𝑤 is
𝑧".
Thus, 𝜑 will ordinarily have free variables
𝑤
and 𝑧- think
of it informally as 𝜑(𝑤, 𝑧). We prefix 𝜑 with the
quantifier ∀𝑦 in order to "protect" the
axiom from any 𝜑
containing 𝑦, thus allowing us to eliminate any
restrictions on
𝜑. Another common variant is derived
as axrep5 5160, where you can
find some further remarks. A slightly more compact version is shown as
axrep2 5157. A quite different variant is zfrep6 7638, which if used in
place of ax-rep 5154 would also require that the Separation Scheme
axsep 5166
be stated as a separate axiom.
There is a very strong generalization of Replacement that doesn't demand function-like behavior of 𝜑. Two versions of this generalization are called the Collection Principle cp 9304 and the Boundedness Axiom bnd 9305. Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5166, Null Set axnul 5173, and Pairing axpr 5294, all of which we derive from Replacement. In order to make it easier to identify the uses of those redundant axioms, we restate them as axioms ax-sep 5167, ax-nul 5174, and ax-pr 5295 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.) |
⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) | ||
Theorem | axrep1 5155* | The version of the Axiom of Replacement used in the Metamath Solitaire applet https://us.metamath.org/mmsolitaire/mms.html. Equivalence is shown via the path ax-rep 5154 → axrep1 5155 → axrep2 5157 → axrepnd 10005 → zfcndrep 10025 = ax-rep 5154. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2379. (Revised by BJ, 31-May-2019.) |
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))) | ||
Theorem | axreplem 5156* | Lemma for axrep2 5157 and axrep3 5158. (Contributed by BJ, 6-Aug-2022.) |
⊢ (𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑥 ∧ 𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑦 ∧ 𝜒))))) | ||
Theorem | axrep2 5157* | Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on 𝜑. (Contributed by NM, 15-Aug-2003.) Remove dependency on ax-13 2379. (Revised by BJ, 31-May-2019.) |
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) | ||
Theorem | axrep3 5158* | Axiom of Replacement slightly strengthened from axrep2 5157; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2379. (Revised by BJ, 31-May-2019.) |
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))) | ||
Theorem | axrep4 5159* | A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))) | ||
Theorem | axrep5 5160* | Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us 𝜑 is analogous to a "function" from 𝑥 to 𝑦 (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set 𝑧 that corresponds to the "image" of 𝜑 restricted to some other set 𝑤. The hypothesis says 𝑧 must not be free in 𝜑. (Contributed by NM, 26-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))) | ||
Theorem | axrep6 5161* | A condensed form of ax-rep 5154. (Contributed by SN, 18-Sep-2023.) |
⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑)) | ||
Theorem | zfrepclf 5162* | An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) ⇒ ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Theorem | zfrep3cl 5163* | An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) ⇒ ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Theorem | zfrep4 5164* | A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.) |
⊢ {𝑥 ∣ 𝜑} ∈ V & ⊢ (𝜑 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) ⇒ ⊢ {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} ∈ V | ||
Theorem | axsepgfromrep 5165* | A more general version axsepg 5168 of the axiom scheme of separation ax-sep 5167 derived from the axiom scheme of replacement ax-rep 5154 (and first-order logic). The extra generality consists in the absence of a disjoint variable condition on 𝑧, 𝜑 (that is, variable 𝑧 may occur in formula 𝜑). See linked statements for more information. (Contributed by NM, 11-Sep-2006.) Remove dependencies on ax-9 2121 to ax-13 2379. (Revised by SN, 25-Sep-2023.) Use ax-sep 5167 instead (or axsepg 5168 if the extra generality is needed). (New usage is discouraged.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Theorem | axsep 5166* | Axiom scheme of separation ax-sep 5167 derived from the axiom scheme of replacement ax-rep 5154. The statement is identical to that of ax-sep 5167, and therefore shows that ax-sep 5167 is redundant when ax-rep 5154 is allowed. See ax-sep 5167 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5167 instead. (New usage is discouraged.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Axiom | ax-sep 5167* |
Axiom scheme of separation. This is an axiom scheme of Zermelo and
Zermelo-Fraenkel set theories.
It was derived as axsep 5166 above and is therefore redundant in ZF set theory, which contains ax-rep 5154 as an axiom (contrary to Zermelo set theory). We state it as a separate axiom here so that some of its uses can be identified more easily. Some textbooks present the axiom scheme of separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger axiom scheme of replacement (which is not part of Zermelo set theory). The axiom scheme of separation is a weak form of Frege's axiom scheme of (unrestricted) comprehension, in that it conditions it with the condition 𝑥 ∈ 𝑧, so that it asserts the existence of a collection only if it is smaller than some other collection 𝑧 that already exists. This prevents Russell's paradox ru 3719. In some texts, this scheme is called "Aussonderung" (German for "separation") or "Subset Axiom". The variable 𝑥 can occur in the formula 𝜑, which in textbooks is often written 𝜑(𝑥). To specify this in the Metamath language, we omit the distinct variable condition ($d) that 𝑥 not occur in 𝜑. For a version using a class variable, see zfauscl 5169, which requires the axiom of extensionality as well as the axiom scheme of separation for its derivation. If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 5227 shows (showing the necessity of that condition in zfauscl 5169). Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Theorem | axsepg 5168* | A more general version of the axiom scheme of separation ax-sep 5167, where variable 𝑧 can also occur (in addition to 𝑥) in formula 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version is derived from the more restrictive ax-sep 5167 with no additional set theory axioms. Note that it was also derived from ax-rep 5154 but without ax-sep 5167 as axsepgfromrep 5165. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2175 and ax-13 2379 and shorten proof. (Revised by BJ, 6-Oct-2019.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Theorem | zfauscl 5169* |
Separation Scheme (Aussonderung) using a class variable. To derive this
from ax-sep 5167, we invoke the Axiom of Extensionality
(indirectly via
vtocl 3507), which is needed for the justification of
class variable
notation.
If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 5227 shows. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Theorem | bm1.3ii 5170* | Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5167. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.) |
⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ⇒ ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) | ||
Theorem | ax6vsep 5171* | Derive ax6v 1971 (a weakened version of ax-6 1970 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5167 and Extensionality ax-ext 2770. See ax6 2391 for the derivation of ax-6 1970 from ax6v 1971. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | axnulALT 5172* | Alternate proof of axnul 5173, proved from propositional calculus, ax-gen 1797, ax-4 1811, sp 2180, and ax-rep 5154. To check this, replace sp 2180 with the obsolete axiom ax-c5 36179 in the proof of axnulALT 5172 and type the Metamath program "MM> SHOW TRACE_BACK axnulALT / AXIOMS" command. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | ||
Theorem | axnul 5173* |
The Null Set Axiom of ZF set theory: there exists a set with no
elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks,
this is presented as a separate axiom; here we show it can be derived
from Separation ax-sep 5167. This version of the Null Set Axiom tells us
that at least one empty set exists, but does not tell us that it is
unique - we need the Axiom of Extensionality to do that (see nulmo 2775).
This proof, suggested by Jeff Hoffman, uses only ax-4 1811 and ax-gen 1797 from predicate calculus, which are valid in "free logic" i.e. logic holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] p. 277). Thus, our ax-sep 5167 implies the existence of at least one set. Note that Kunen's version of ax-sep 5167 (Axiom 3 of [Kunen] p. 11) does not imply the existence of a set because his is universally closed, i.e., prefixed with universal quantifiers to eliminate all free variables. His existence is provided by a separate axiom stating ∃𝑥𝑥 = 𝑥 (Axiom 0 of [Kunen] p. 10). See axnulALT 5172 for a proof directly from ax-rep 5154. This theorem should not be referenced by any proof. Instead, use ax-nul 5174 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | ||
Axiom | ax-nul 5174* | The Null Set Axiom of ZF set theory. It was derived as axnul 5173 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 7-Aug-2003.) |
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | ||
Theorem | 0ex 5175 | The Null Set Axiom of ZF set theory: the empty set exists. Corollary 5.16 of [TakeutiZaring] p. 20. For the unabbreviated version, see ax-nul 5174. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ∅ ∈ V | ||
Theorem | al0ssb 5176* | The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.) |
⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅) | ||
Theorem | sseliALT 5177 | Alternate proof of sseli 3911 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3912. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) | ||
Theorem | csbexg 5178 | The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.) |
⊢ (∀𝑥 𝐵 ∈ 𝑊 → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | ||
Theorem | csbex 5179 | The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Revised by NM, 17-Aug-2018.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V | ||
Theorem | unisn2 5180 | A version of unisn 4820 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.) |
⊢ ∪ {𝐴} ∈ {∅, 𝐴} | ||
Theorem | nalset 5181* | No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2175 and ax-13 2379. (Revised by BJ, 31-May-2019.) |
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | ||
Theorem | vnex 5182 | The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.) |
⊢ ¬ ∃𝑥 𝑥 = V | ||
Theorem | vprc 5183 | The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.) |
⊢ ¬ V ∈ V | ||
Theorem | nvel 5184 | The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) |
⊢ ¬ V ∈ 𝐴 | ||
Theorem | inex1 5185 | Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∩ 𝐵) ∈ V | ||
Theorem | inex2 5186 | Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∩ 𝐴) ∈ V | ||
Theorem | inex1g 5187 | Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | inex2g 5188 | Sufficient condition for an intersection to be a set. Commuted form of inex1g 5187. (Contributed by Peter Mazsa, 19-Dec-2018.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) | ||
Theorem | ssex 5189 | The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 5167 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) | ||
Theorem | ssexi 5190 | The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.) |
⊢ 𝐵 ∈ V & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | ssexg 5191 | The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | ||
Theorem | ssexd 5192 | A subclass of a set is a set. Deduction form of ssexg 5191. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | prcssprc 5193 | The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) | ||
Theorem | sselpwd 5194 | Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | difexg 5195 | Existence of a difference. (Contributed by NM, 26-May-1998.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) | ||
Theorem | difexi 5196 | Existence of a difference, inference version of difexg 5195. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∖ 𝐵) ∈ V | ||
Theorem | zfausab 5197* | Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V | ||
Theorem | rabexg 5198* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | rabex 5199* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V | ||
Theorem | rabexd 5200* | Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5201. (Contributed by AV, 16-Jul-2019.) |
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 ∈ V) |
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