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Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembreqtrid 4001 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴𝑅𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrrid 4002 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
𝐴𝑅𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrdi 4003 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrrdi 4004 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrdi 4005 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrrdi 4006 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝑅𝐶)
 
Theoremssbrd 4007 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
 
Theoremssbri 4008 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
𝐴𝐵       (𝐶𝐴𝐷𝐶𝐵𝐷)
 
Theoremnfbrd 4009 Deduction version of bound-variable hypothesis builder nfbr 4010. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝑅)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
 
Theoremnfbr 4010 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝐴    &   𝑥𝑅    &   𝑥𝐵       𝑥 𝐴𝑅𝐵
 
Theorembrab1 4011* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
(𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
 
Theorembr0 4012 The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
¬ 𝐴𝐵
 
Theorembrne0 4013 If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4014. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
(𝐴𝑅𝐵𝑅 ≠ ∅)
 
Theorembrm 4014* If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
(𝐴𝑅𝐵 → ∃𝑥 𝑥𝑅)
 
Theorembrun 4015 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembrin 4016 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembrdif 4017 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))
 
Theoremsbcbrg 4018 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
 
Theoremsbcbr12g 4019* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶))
 
Theoremsbcbr1g 4020* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))
 
Theoremsbcbr2g 4021* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝑅𝐴 / 𝑥𝐶))
 
Theorembrralrspcev 4022* 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 4023* 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))
 
2.1.23  Ordered-pair class abstractions (class builders)
 
Syntaxcopab 4024 Extend class notation to include ordered-pair class abstraction (class builder).
class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Syntaxcmpt 4025 Extend the definition of a class to include maps-to notation for defining a function via a rule.
class (𝑥𝐴𝐵)
 
Definitiondf-opab 4026* 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 doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
 
Definitiondf-mpt 4027* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from 𝑥 (in 𝐴) to 𝐵(𝑥)." The class expression 𝐵 is the value of the function at 𝑥 and normally contains the variable 𝑥. Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)
(𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
 
Theoremopabss 4028* 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 4029 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 4030* 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 4031 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
(𝜑𝜓)       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
 
Theoremnfopab 4032* 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 4033 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremnfopab2 4034 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremcbvopab 4035* 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 4036* 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 4037* 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.)
𝑧𝜑    &   𝑥𝜓    &   (𝑥 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
 
Theoremcbvopab2 4038* 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 4039* 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 4040* 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 4041* 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}
 
Theoremcsbopabg 4042* Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
 
Theoremunopab 4043 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}
 
Theoremmpteq12f 4044 An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq12dva 4045* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq12dv 4046* An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq12 4047* An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)
((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq1 4048* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 
Theoremmpteq1d 4049* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 
Theoremmpteq2ia 4050 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝑥𝐴𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
 
Theoremmpteq2i 4051 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
𝐵 = 𝐶       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
 
Theoremmpteq12i 4052 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 4053 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 4054* Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
 
Theoremmpteq2dv 4055* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
 
Theoremnfmpt 4056* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝑦𝐴𝐵)
 
Theoremnfmpt1 4057 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
𝑥(𝑥𝐴𝐵)
 
Theoremcbvmptf 4058* 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 Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
Theoremcbvmpt 4059* 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.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
Theoremcbvmptv 4060* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
(𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
Theoremmptv 4061* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
(𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}
 
2.1.24  Transitive classes
 
Syntaxwtr 4062 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴
 
Definitiondf-tr 4063 Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4064 (which is suggestive of the word "transitive"), dftr3 4066, dftr4 4067, and dftr5 4065. 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 4064* 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 4065* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
(Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
 
Theoremdftr3 4066* 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 4067 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
 
Theoremtreq 4068 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
 
Theoremtrel 4069 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 4070 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
(Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))
 
Theoremtrss 4071 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
(Tr 𝐴 → (𝐵𝐴𝐵𝐴))
 
Theoremtrin 4072 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
 
Theoremtr0 4073 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Tr ∅
 
Theoremtrv 4074 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
Tr V
 
Theoremtriun 4075* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
(∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
 
Theoremtruni 4076* 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)
 
Theoremtrint 4077* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 
Theoremtrintssm 4078* Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)
((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)
 
2.2  IZF Set Theory - add the Axioms of Collection and Separation
 
2.2.1  Introduce the Axiom of Collection
 
Axiomax-coll 4079* Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4133 but uses a freeness hypothesis in place of one of the distinct variable conditions. (Contributed by Jim Kingdon, 23-Aug-2018.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
 
Theoremrepizf 4080* Axiom of Replacement. Axiom 7' of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). In our context this is not an axiom, but a theorem proved from ax-coll 4079. It is identical to zfrep6 4081 except for the choice of a freeness hypothesis rather than a disjoint variable condition between 𝑏 and 𝜑. (Contributed by Jim Kingdon, 23-Aug-2018.)
𝑏𝜑       (∀𝑥𝑎 ∃!𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
 
Theoremzfrep6 4081* A version of the Axiom of Replacement. Normally 𝜑 would have free variables 𝑥 and 𝑦. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4082 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version. (Contributed by NM, 10-Oct-2003.)
(∀𝑥𝑧 ∃!𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)
 
2.2.2  Introduce the Axiom of Separation
 
Axiomax-sep 4082* The Axiom of Separation of IZF set theory. Axiom 6 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed, and with a 𝑦𝜑 condition replaced by a disjoint variable condition between 𝑦 and 𝜑).

The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with 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 2936. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

(Contributed by NM, 11-Sep-2006.)

𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
Theoremaxsep2 4083* A less restrictive version of the Separation Scheme ax-sep 4082, where variables 𝑥 and 𝑧 can both appear free in the wff 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version was derived from the more restrictive ax-sep 4082 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
Theoremzfauscl 4084* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4082, we invoke the Axiom of Extensionality (indirectly via vtocl 2766), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
 
Theorembm1.3ii 4085* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4082. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)
𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)
 
Theorema9evsep 4086* Derive a weakened version of ax-i9 1510, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4082 and Extensionality ax-ext 2139. The theorem ¬ ∀𝑥¬ 𝑥 = 𝑦 also holds (ax9vsep 4087), but in intuitionistic logic 𝑥𝑥 = 𝑦 is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦
 
Theoremax9vsep 4087* Derive a weakened version of ax-9 1511, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4082 and Extensionality ax-ext 2139. In intuitionistic logic a9evsep 4086 is stronger and also holds. (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
 
Theoremzfnuleu 4088* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2142 to strengthen the hypothesis in the form of axnul 4089). (Contributed by NM, 22-Dec-2007.)
𝑥𝑦 ¬ 𝑦𝑥       ∃!𝑥𝑦 ¬ 𝑦𝑥
 
Theoremaxnul 4089* 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 4082. 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 zfnuleu 4088).

This theorem should not be referenced by any proof. Instead, use ax-nul 4090 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 4090* The Null Set Axiom of IZF set theory. It was derived as axnul 4089 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. Axiom 4 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by NM, 7-Aug-2003.)
𝑥𝑦 ¬ 𝑦𝑥
 
Theorem0ex 4091 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 4090. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
∅ ∈ V
 
Theoremcsbexga 4092 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
 
Theoremcsbexa 4093 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴 / 𝑥𝐵 ∈ V
 
2.2.4  Theorems requiring subset and intersection existence
 
Theoremnalset 4094* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
¬ ∃𝑥𝑦 𝑦𝑥
 
Theoremvnex 4095 The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
¬ ∃𝑥 𝑥 = V
 
Theoremvprc 4096 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
 
Theoremnvel 4097 The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
¬ V ∈ 𝐴
 
Theoreminex1 4098 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theoreminex2 4099 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
𝐴 ∈ V       (𝐵𝐴) ∈ V
 
Theoreminex1g 4100 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)
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