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Theorem List for Metamath Proof Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembr0 5101 The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
¬ 𝐴𝐵

Theorembrne0 5102 If two sets are in a binary relation, the relation cannot be empty. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
(𝐴𝑅𝐵𝑅 ≠ ∅)

Theorembrun 5103 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))

Theorembrin 5104 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))

Theorembrdif 5105 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))

Theoremsbcbr123 5106 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 22-Aug-2018.)
([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)

Theoremsbcbr 5107* Move substitution in and out of a binary relation. (Contributed by NM, 23-Aug-2018.)
([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝐴 / 𝑥𝑅𝐶)

Theoremsbcbr12g 5108* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶))

Theoremsbcbr1g 5109* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))

Theoremsbcbr2g 5110* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝑅𝐴 / 𝑥𝐶))

Theorembrsymdif 5111 Characterization of the symmetric difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2012.)
(𝐴(𝑅𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵𝐴𝑆𝐵))

Theorembrralrspcev 5112* Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
((𝐵𝑋 ∧ ∀𝑦𝑌 𝐴𝑅𝐵) → ∃𝑥𝑋𝑦𝑌 𝐴𝑅𝑥)

Theorembrimralrspcev 5113* Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)
((𝐵𝑋 ∧ ∀𝑦𝑌 ((𝜑𝐴𝑅𝐵) → 𝜓)) → ∃𝑥𝑋𝑦𝑌 ((𝜑𝐴𝑅𝑥) → 𝜓))

2.1.24  Ordered-pair class abstractions (class builders)

Syntaxcopab 5114 Extend class notation to include ordered-pair class abstraction (class builder).
class {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Definitiondf-opab 5115* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually 𝑥 and 𝑦 are distinct, although the definition does not require it (see dfid2 5450 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also called "class builder" in the literature. An alternate definition using no existential quantifiers is shown by dfopab2 7745. An example is given by ex-opab 28220. (Contributed by NM, 4-Jul-1994.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

Theoremopabss 5116* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅

Theoremopabbid 5117 Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Theoremopabbidv 5118* Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Theoremopabbii 5119 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
(𝜑𝜓)       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Theoremnfopab 5120* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.)
𝑧𝜑       𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Theoremnfopab1 5121 The first abstraction variable in an ordered-pair class abstraction is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Theoremnfopab2 5122 The second abstraction variable in an ordered-pair class abstraction is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Theoremcbvopab 5123* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

Theoremcbvopabv 5124* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

Theoremcbvopab1 5125* 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 2392. See cbvopab1g 5126 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
𝑧𝜑    &   𝑥𝜓    &   (𝑥 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

Theoremcbvopab1g 5126* 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 2392. See cbvopab1 5125 for a version with more disjoint variable conditions, but not requiring ax-13 2392. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
𝑧𝜑    &   𝑥𝜓    &   (𝑥 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

Theoremcbvopab2 5127* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
𝑧𝜑    &   𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}

Theoremcbvopab1s 5128* 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}

Theoremcbvopab1v 5129* 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 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

Theoremcbvopab2v 5130* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
(𝑦 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}

Theoremunopab 5131 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}

2.1.25  Functions in maps-to notation

Syntaxcmpt 5132 Extend the definition of a class to include maps-to notation for defining a function via a rule.
class (𝑥𝐴𝐵)

Definitiondf-mpt 5133* 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.)
(𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}

Theoremmpteq12df 5134 An equality inference for the maps-to notation. Compare mpteq12dv 5137. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12f 5135 An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12dva 5136* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12dv 5137* An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) Drop ax-10 2146 while shortening its proof. (Revised by Steven Nguyen and Gino Giotto, 1-Dec-2023.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12dvOLD 5138* Obsolete version of mpteq12dv 5137 as of 1-Dec-2023. (Contributed by NM, 24-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq12 5139* An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)
((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremmpteq1 5140* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))

Theoremmpteq1d 5141* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))

Theoremmpteq1i 5142* An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐴 = 𝐵       (𝑥𝐴𝐶) = (𝑥𝐵𝐶)

Theoremmpteq2ia 5143 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝑥𝐴𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)

Theoremmpteq2i 5144 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
𝐵 = 𝐶       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)

Theoremmpteq12i 5145 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       (𝑥𝐴𝐵) = (𝑥𝐶𝐷)

Theoremmpteq2da 5146 Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))

Theoremmpteq2dva 5147* Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))

Theoremmpteq2dv 5148* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))

Theoremnfmpt 5149* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝑦𝐴𝐵)

Theoremnfmpt1 5150 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
𝑥(𝑥𝐴𝐵)

Theoremcbvmptf 5151* 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 2392. See cbvmptfg 5152 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Theoremcbvmptfg 5152 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 2392. See cbvmptf 5151 for a version with more disjoint variable conditions, but not requiring ax-13 2392. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) (New usage is discouraged.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Theoremcbvmpt 5153* 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 2392. See cbvmptg 5154 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Theoremcbvmptg 5154* 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 2392. See cbvmpt 5153 for a version with more disjoint variable conditions, but not requiring ax-13 2392. (Contributed by NM, 11-Sep-2011.) (New usage is discouraged.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Theoremcbvmptv 5155* 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 2392. See cbvmptvg 5156 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
(𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Theoremcbvmptvg 5156* 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 2392. See cbvmptv 5155 for a version with more disjoint variable conditions, but not requiring ax-13 2392. (Contributed by Mario Carneiro, 19-Feb-2013.) (New usage is discouraged.)
(𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Theoremmptv 5157* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
(𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}

2.1.26  Transitive classes

Syntaxwtr 5158 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴

Definitiondf-tr 5159 Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5959). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 5160 (which is suggestive of the word "transitive"), dftr3 5162, dftr4 5163, dftr5 5161, and (when 𝐴 is a set) unisuc 6254. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴 𝐴𝐴)

Theoremdftr2 5160* An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)
(Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))

Theoremdftr5 5161* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
(Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)

Theoremdftr3 5162* An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)

Theoremdftr4 5163 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Theoremtreq 5164 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))

Theoremtrel 5165 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 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))

Theoremtrel3 5166 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
(Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))

Theoremtrss 5167 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 → (𝐵𝐴𝐵𝐴))

Theoremtrin 5168 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))

Theoremtr0 5169 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Tr ∅

Theoremtrv 5170 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
Tr V

Theoremtriun 5171 An indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
(∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)

Theoremtruni 5172* 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.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

Theoremtriin 5173 An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.)
(∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)

Theoremtrint 5174* 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.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

Theoremtrintss 5175 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 𝐴𝐴 ≠ ∅) → 𝐴𝐴)

2.2  ZF Set Theory - add the Axiom of Replacement

2.2.1  Introduce the Axiom of Replacement

Axiomax-rep 5176* 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 6429). 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 5182, where you can find some further remarks. A slightly more compact version is shown as axrep2 5179. A quite different variant is zfrep6 7651, which if used in place of ax-rep 5176 would also require that the Separation Scheme axsep 5188 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 9317 and the Boundedness Axiom bnd 9318.

Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5188, Null Set axnul 5195, and Pairing axpr 5316, 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 5189, ax-nul 5196, and ax-pr 5317 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)

(∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))

Theoremaxrep1 5177* 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 5176 axrep1 5177 axrep2 5179 axrepnd 10014 zfcndrep 10034 = ax-rep 5176. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2392. (Revised by BJ, 31-May-2019.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))

Theoremaxreplem 5178* Lemma for axrep2 5179 and axrep3 5180. (Contributed by BJ, 6-Aug-2022.)
(𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑥𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑦𝜒)))))

Theoremaxrep2 5179* 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 2392. (Revised by BJ, 31-May-2019.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))

Theoremaxrep3 5180* Axiom of Replacement slightly strengthened from axrep2 5179; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2392. (Revised by BJ, 31-May-2019.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))

Theoremaxrep4 5181* A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
𝑧𝜑       (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theoremaxrep5 5182* 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.)
𝑧𝜑       (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))

Theoremaxrep6 5183* A condensed form of ax-rep 5176. (Contributed by SN, 18-Sep-2023.)
(∀𝑤∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))

Theoremzfrepclf 5184* An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
𝑥𝐴    &   𝐴 ∈ V    &   (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))       𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))

Theoremzfrep3cl 5185* 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    &   (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))       𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))

Theoremzfrep4 5186* A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)
{𝑥𝜑} ∈ V    &   (𝜑 → ∃𝑧𝑦(𝜓𝑦 = 𝑧))       {𝑦 ∣ ∃𝑥(𝜑𝜓)} ∈ V

2.2.2  Derive the Axiom of Separation

Theoremaxsepgfromrep 5187* A more general version axsepg 5190 of the axiom scheme of separation ax-sep 5189 derived from the axiom scheme of replacement ax-rep 5176 (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 2125 to ax-13 2392. (Revised by SN, 25-Sep-2023.) Use ax-sep 5189 instead (or axsepg 5190 if the extra generality is needed). (New usage is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theoremaxsep 5188* Axiom scheme of separation ax-sep 5189 derived from the axiom scheme of replacement ax-rep 5176. The statement is identical to that of ax-sep 5189, and therefore shows that ax-sep 5189 is redundant when ax-rep 5176 is allowed. See ax-sep 5189 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5189 instead. (New usage is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Axiomax-sep 5189* Axiom scheme of separation. This is an axiom scheme of Zermelo and Zermelo-Fraenkel set theories.

It was derived as axsep 5188 above and is therefore redundant in ZF set theory, which contains ax-rep 5176 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 𝑥𝑧, 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 3757. 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 5191, 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 5249 shows (showing the necessity of that condition in zfauscl 5191).

Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.)

𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theoremaxsepg 5190* A more general version of the axiom scheme of separation ax-sep 5189, 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 5189 with no additional set theory axioms. Note that it was also derived from ax-rep 5176 but without ax-sep 5189 as axsepgfromrep 5187. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2179 and ax-13 2392 and shorten proof. (Revised by BJ, 6-Oct-2019.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theoremzfauscl 5191* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 5189, we invoke the Axiom of Extensionality (indirectly via vtocl 3545), 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 5249 shows. (Contributed by NM, 21-Jun-1993.)

𝐴 ∈ V       𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

Theorembm1.3ii 5192* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5189. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)
𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)

Theoremax6vsep 5193* Derive ax6v 1972 (a weakened version of ax-6 1971 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5189 and Extensionality ax-ext 2796. See ax6 2404 for the derivation of ax-6 1971 from ax6v 1972. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦

2.2.3  Derive the Null Set Axiom

TheoremaxnulALT 5194* Alternate proof of axnul 5195, proved from propositional calculus, ax-gen 1797, ax-4 1811, sp 2184, and ax-rep 5176. To check this, replace sp 2184 with the obsolete axiom ax-c5 36124 in the proof of axnulALT 5194 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.)
𝑥𝑦 ¬ 𝑦𝑥

Theoremaxnul 5195* 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 5189. 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 2801).

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 5189 implies the existence of at least one set. Note that Kunen's version of ax-sep 5189 (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 5194 for a proof directly from ax-rep 5176.

This theorem should not be referenced by any proof. Instead, use ax-nul 5196 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.)

𝑥𝑦 ¬ 𝑦𝑥

Axiomax-nul 5196* The Null Set Axiom of ZF set theory. It was derived as axnul 5195 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.)
𝑥𝑦 ¬ 𝑦𝑥

Theorem0ex 5197 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 5196. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
∅ ∈ V

Theoremal0ssb 5198* The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.)
(∀𝑦 𝑋𝑦𝑋 = ∅)

TheoremsseliALT 5199 Alternate proof of sseli 3949 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3950. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)

Theoremcsbexg 5200 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.)
(∀𝑥 𝐵𝑊𝐴 / 𝑥𝐵 ∈ V)

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