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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mpteq2ia 3901 | 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 3902 | 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 3903 | 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 3904 | 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 3905* | 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 3906* | 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 3907* | Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | nfmpt1 3908 | Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.) |
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | cbvmptf 3909* | 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.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmpt 3910* | 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.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | cbvmptv 3911* | Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) | ||
Theorem | mptv 3912* | Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
⊢ (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵} | ||
Syntax | wtr 3913 | Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35. |
wff Tr 𝐴 | ||
Definition | df-tr 3914 | Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 3915 (which is suggestive of the word "transitive"), dftr3 3917, dftr4 3918, and dftr5 3916. 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 3915* | 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 3916* | An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.) |
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | ||
Theorem | dftr3 3917* | 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 3918 | 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 3919 | Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.) |
⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵)) | ||
Theorem | trel 3920 | 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 3921 | 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 3922 | An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) |
⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | ||
Theorem | trin 3923 | The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.) |
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) | ||
Theorem | tr0 3924 | The empty set is transitive. (Contributed by NM, 16-Sep-1993.) |
⊢ Tr ∅ | ||
Theorem | trv 3925 | The universe is transitive. (Contributed by NM, 14-Sep-2003.) |
⊢ Tr V | ||
Theorem | triun 3926* | The 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 3927* | 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 | trint 3928* | The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.) |
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴) | ||
Theorem | trintssm 3929* | 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 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ 𝐴) | ||
Theorem | trintssmOLD 3930* | Obsolete version of trintssm 3929 as of 30-Oct-2021. (Contributed by Jim Kingdon, 22-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ Tr 𝐴) → ∩ 𝐴 ⊆ 𝐴) | ||
Axiom | ax-coll 3931* | Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3984 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.) |
⊢ Ⅎ𝑏𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑) | ||
Theorem | repizf 3932* | 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 3931. It is identical to zfrep6 3933 except for the choice of a freeness hypothesis rather than a distinct variable constraint between 𝑏 and 𝜑. (Contributed by Jim Kingdon, 23-Aug-2018.) |
⊢ Ⅎ𝑏𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝑎 ∃!𝑦𝜑 → ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑) | ||
Theorem | zfrep6 3933* | 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 3934 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.) |
⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑) | ||
Axiom | ax-sep 3934* |
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
Ⅎ𝑦𝜑 condition replaced by a distinct
variable constraint between
𝑦 and 𝜑).
The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with 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 2828. In some texts, this scheme is called "Aussonderung" or the Subset Axiom. (Contributed by NM, 11-Sep-2006.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Theorem | axsep2 3935* | A less restrictive version of the Separation Scheme ax-sep 3934, 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 3934 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) | ||
Theorem | zfauscl 3936* | Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3934, we invoke the Axiom of Extensionality (indirectly via vtocl 2667), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
Theorem | bm1.3ii 3937* | Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3934. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.) |
⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ⇒ ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) | ||
Theorem | a9evsep 3938* | Derive a weakened version of ax-i9 1466, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 3934 and Extensionality ax-ext 2067. The theorem ¬ ∀𝑥¬ 𝑥 = 𝑦 also holds (ax9vsep 3939), 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 | ||
Theorem | ax9vsep 3939* | Derive a weakened version of ax-9 1467, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 3934 and Extensionality ax-ext 2067. In intuitionistic logic a9evsep 3938 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | zfnuleu 3940* | Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2070 to strengthen the hypothesis in the form of axnul 3941). (Contributed by NM, 22-Dec-2007.) |
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ⇒ ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | ||
Theorem | axnul 3941* |
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 3934. 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 3940).
This theorem should not be referenced by any proof. Instead, use ax-nul 3942 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 3942* | The Null Set Axiom of IZF set theory. It was derived as axnul 3941 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.) |
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | ||
Theorem | 0ex 3943 | 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 3942. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ∅ ∈ V | ||
Theorem | csbexga 3944 | The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 𝐵 ∈ 𝑊) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | ||
Theorem | csbexa 3945 | 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 / 𝑥⦌𝐵 ∈ V | ||
Theorem | nalset 3946* | No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) |
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | ||
Theorem | vnex 3947 | The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.) |
⊢ ¬ ∃𝑥 𝑥 = V | ||
Theorem | vprc 3948 | 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 3949 | The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) |
⊢ ¬ V ∈ 𝐴 | ||
Theorem | inex1 3950 | Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∩ 𝐵) ∈ V | ||
Theorem | inex2 3951 | Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∩ 𝐴) ∈ V | ||
Theorem | inex1g 3952 | Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | ssex 3953 | 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 3934 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) | ||
Theorem | ssexi 3954 | The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.) |
⊢ 𝐵 ∈ V & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | ssexg 3955 | 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 3956 | A subclass of a set is a set. Deduction form of ssexg 3955. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | difexg 3957 | Existence of a difference. (Contributed by NM, 26-May-1998.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) | ||
Theorem | zfausab 3958* | Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V | ||
Theorem | rabexg 3959* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | rabex 3960* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V | ||
Theorem | elssabg 3961* | Membership in a class abstraction involving a subset. Unlike elabg 2752, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝜑)} ↔ (𝐴 ⊆ 𝐵 ∧ 𝜓))) | ||
Theorem | inteximm 3962* | The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) | ||
Theorem | intexr 3963 | If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.) |
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) | ||
Theorem | intnexr 3964 | If a class intersection is the universe, it is not a set. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 27-Aug-2018.) |
⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) | ||
Theorem | intexabim 3965 | The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
⊢ (∃𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} ∈ V) | ||
Theorem | intexrabim 3966 | The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | iinexgm 3967* | The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Jim Kingdon, 28-Aug-2018.) |
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) | ||
Theorem | inuni 3968* | The intersection of a union ∪ 𝐴 with a class 𝐵 is equal to the union of the intersections of each element of 𝐴 with 𝐵. (Contributed by FL, 24-Mar-2007.) |
⊢ (∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)} | ||
Theorem | elpw2g 3969 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpw2 3970 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) | ||
Theorem | pwnss 3971 | The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴) | ||
Theorem | pwne 3972 | No set equals its power set. The sethood antecedent is necessary; compare pwv 3637. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴) | ||
Theorem | repizf2lem 3973 | Lemma for repizf2 3974. If we have a function-like proposition which provides at most one value of 𝑦 for each 𝑥 in a set 𝑤, we can change "at most one" to "exactly one" by restricting the values of 𝑥 to those values for which the proposition provides a value of 𝑦. (Contributed by Jim Kingdon, 7-Sep-2018.) |
⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑) | ||
Theorem | repizf2 3974* | Replacement. This version of replacement is stronger than repizf 3932 in the sense that 𝜑 does not need to map all values of 𝑥 in 𝑤 to a value of 𝑦. The resulting set contains those elements for which there is a value of 𝑦 and in that sense, this theorem combines repizf 3932 with ax-sep 3934. Another variation would be ∀𝑥 ∈ 𝑤∃*𝑦𝜑 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)} ∈ V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 → ∃𝑧∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑) | ||
Theorem | class2seteq 3975* | Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) | ||
Theorem | 0elpw 3976 | Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.) |
⊢ ∅ ∈ 𝒫 𝐴 | ||
Theorem | 0nep0 3977 | The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.) |
⊢ ∅ ≠ {∅} | ||
Theorem | 0inp0 3978 | Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.) |
⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅}) | ||
Theorem | unidif0 3979 | The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) |
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴 | ||
Theorem | iin0imm 3980* | An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 ∅ = ∅) | ||
Theorem | iin0r 3981* | If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅) | ||
Theorem | intv 3982 | The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.) |
⊢ ∩ V = ∅ | ||
Theorem | axpweq 3983* | Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3986 is not used by the proof. (Contributed by NM, 22-Jun-2009.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥)) | ||
Theorem | bnd 3984* | A very strong generalization of the Axiom of Replacement (compare zfrep6 3933). Its strength lies in the rather profound fact that 𝜑(𝑥, 𝑦) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. In the context of IZF, it is just a slight variation of ax-coll 3931. (Contributed by NM, 17-Oct-2004.) |
⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑) | ||
Theorem | bnd2 3985* | A variant of the Boundedness Axiom bnd 3984 that picks a subset 𝑧 out of a possibly proper class 𝐵 in which a property is true. (Contributed by NM, 4-Feb-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑)) | ||
Axiom | ax-pow 3986* |
Axiom of Power Sets. An axiom of Intuitionistic Zermelo-Fraenkel set
theory. It states that a set 𝑦 exists that includes the power set
of a given set 𝑥 i.e. contains every subset of 𝑥. This
is
Axiom 8 of [Crosilla] p. "Axioms
of CZF and IZF" except (a) unnecessary
quantifiers are removed, and (b) Crosilla has a biconditional rather
than an implication (but the two are equivalent by bm1.3ii 3937).
The variant axpow2 3988 uses explicit subset notation. A version using class notation is pwex 3994. (Contributed by NM, 5-Aug-1993.) |
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
Theorem | zfpow 3987* | Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) |
⊢ ∃𝑥∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
Theorem | axpow2 3988* | A variant of the Axiom of Power Sets ax-pow 3986 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) | ||
Theorem | axpow3 3989* | A variant of the Axiom of Power Sets ax-pow 3986. For any set 𝑥, there exists a set 𝑦 whose members are exactly the subsets of 𝑥 i.e. the power set of 𝑥. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) | ||
Theorem | el 3990* | Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ ∃𝑦 𝑥 ∈ 𝑦 | ||
Theorem | vpwex 3991 | Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 3992 from vpwex 3991. (Revised by BJ, 10-Aug-2022.) |
⊢ 𝒫 𝑥 ∈ V | ||
Theorem | pwexg 3992 | Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | ||
Theorem | abssexg 3993* | Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V) | ||
Theorem | pwex 3994 | Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝒫 𝐴 ∈ V | ||
Theorem | snexg 3995 | A singleton whose element exists is a set. The 𝐴 ∈ V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | ||
Theorem | snex 3996 | A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝐴} ∈ V | ||
Theorem | snexprc 3997 | A singleton whose element is a proper class is a set. The ¬ 𝐴 ∈ V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.) |
⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) | ||
Theorem | notnotsnex 3998 | A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.) |
⊢ ¬ ¬ {𝐴} ∈ V | ||
Theorem | p0ex 3999 | The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.) |
⊢ {∅} ∈ V | ||
Theorem | pp0ex 4000 | {∅, {∅}} (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.) |
⊢ {∅, {∅}} ∈ V |
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