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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cbvopab2 3801* | 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 3802* | 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 3803* | 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 3804* | 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 | csbopabg 3805* | Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
⊢ (A ∈ 𝑉 → ⦋A / x⦌{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [A / x]φ}) | ||
Theorem | unopab 3806 | Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.) |
⊢ ({⟨x, y⟩ ∣ φ} ∪ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ∨ ψ)} | ||
Theorem | mpteq12f 3807 | 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 3808* | 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 3809* | An equality inference for the maps to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) |
⊢ (φ → A = 𝐶) & ⊢ (φ → B = 𝐷) ⇒ ⊢ (φ → (x ∈ A ↦ B) = (x ∈ 𝐶 ↦ 𝐷)) | ||
Theorem | mpteq12 3810* | 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 3811* | 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 3812* | 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 | mpteq2ia 3813 | 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 3814 | 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 3815 | 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 3816 | 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 3817* | 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 3818* | 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 3819* | Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
⊢ ℲxA & ⊢ ℲxB ⇒ ⊢ Ⅎx(y ∈ A ↦ B) | ||
Theorem | nfmpt1 3820 | Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.) |
⊢ Ⅎx(x ∈ A ↦ B) | ||
Theorem | cbvmpt 3821* | 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 3822* | 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 3823* | Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
⊢ (x ∈ V ↦ B) = {⟨x, y⟩ ∣ y = B} | ||
Syntax | wtr 3824 | Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35. |
wff Tr A | ||
Definition | df-tr 3825 | Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 3826 (which is suggestive of the word "transitive"), dftr3 3828, dftr4 3829, and dftr5 3827. 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 3826* | 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 3827* | An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.) |
⊢ (Tr A ↔ ∀x ∈ A ∀y ∈ x y ∈ A) | ||
Theorem | dftr3 3828* | 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 3829 | 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 3830 | Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.) |
⊢ (A = B → (Tr A ↔ Tr B)) | ||
Theorem | trel 3831 | 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 3832 | 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 3833 | 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 3834 | The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.) |
⊢ ((Tr A ∧ Tr B) → Tr (A ∩ B)) | ||
Theorem | tr0 3835 | The empty set is transitive. (Contributed by NM, 16-Sep-1993.) |
⊢ Tr ∅ | ||
Theorem | trv 3836 | The universe is transitive. (Contributed by NM, 14-Sep-2003.) |
⊢ Tr V | ||
Theorem | triun 3837* | 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 3838* | 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 3839* | 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 3840* | If A is transitive and inhabited, then ∩ A is a subset of A. (Contributed by Jim Kingdon, 22-Aug-2018.) |
⊢ ((∃x x ∈ A ∧ Tr A) → ∩ A ⊆ A) | ||
Theorem | trint0m 3841* | Any inhabited transitive class includes its intersection. Similar to Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Jim Kingdon, 22-Aug-2018.) |
⊢ ((Tr A ∧ ∃x x ∈ A) → ∩ A ⊆ A) | ||
Axiom | ax-coll 3842* | Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3895 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 3843* | 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 3842. It is identical to zfrep6 3844 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 3844* | 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 3845 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 3845* |
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 2736. 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 3846* | A less restrictive version of the Separation Scheme ax-sep 3845, 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 3845 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 3847* | Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3845, we invoke the Axiom of Extensionality (indirectly via vtocl 2581), 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 3848* | Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3845. 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 3849* | Derive a weakened version of ax-i9 1400, where x and y must be distinct, from Separation ax-sep 3845 and Extensionality ax-ext 2000. The theorem ¬ ∀x¬ x = y also holds (ax9vsep 3850), 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 3850* | Derive a weakened version of ax-9 1401, where x and y must be distinct, from Separation ax-sep 3845 and Extensionality ax-ext 2000. In intuitionistic logic a9evsep 3849 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∀x ¬ x = y | ||
Theorem | zfnuleu 3851* | Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2003 to strengthen the hypothesis in the form of axnul 3852). (Contributed by NM, 22-Dec-2007.) |
⊢ ∃x∀y ¬ y ∈ x ⇒ ⊢ ∃!x∀y ¬ y ∈ x | ||
Theorem | axnul 3852* |
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 3845. 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 3851).
This theorem should not be referenced by any proof. Instead, use ax-nul 3853 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 3853* | The Null Set Axiom of IZF set theory. It was derived as axnul 3852 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 3854 | 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 3853. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ∅ ∈ V | ||
Theorem | csbexgOLD 3855 | The existence of proper substitution into a class. Although we adopt the name csbexgOLD 3855 for consistency with the Metamath Proof Explorer, we do not yet have a version of this theorem without the A ∈ 𝑉 condition, so unless/until that happens, this theorem is not obsolete and can be used. (Contributed by NM, 10-Nov-2005.) |
⊢ ((A ∈ 𝑉 ∧ ∀x B ∈ 𝑊) → ⦋A / x⦌B ∈ V) | ||
Theorem | csbexOLD 3856 | The existence of proper substitution into a class. Although we adopt the name csbexOLD 3856 for consistency with the Metamath Proof Explorer, we do not yet have a version of this theorem without the A ∈ 𝑉 condition, so unless/until that happens, this theorem is not obsolete and can be used. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ ⦋A / x⦌B ∈ V | ||
Theorem | nalset 3857* | No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) |
⊢ ¬ ∃x∀y y ∈ x | ||
Theorem | vprc 3858 | 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 3859 | The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.) |
⊢ ¬ V ∈ A | ||
Theorem | vnex 3860 | The universal class does not exist. (Contributed by NM, 4-Jul-2005.) |
⊢ ¬ ∃x x = V | ||
Theorem | inex1 3861 | 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 3862 | Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.) |
⊢ A ∈ V ⇒ ⊢ (B ∩ A) ∈ V | ||
Theorem | inex1g 3863 | Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.) |
⊢ (A ∈ 𝑉 → (A ∩ B) ∈ V) | ||
Theorem | ssex 3864 | 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 3845 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) |
⊢ B ∈ V ⇒ ⊢ (A ⊆ B → A ∈ V) | ||
Theorem | ssexi 3865 | The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.) |
⊢ B ∈ V & ⊢ A ⊆ B ⇒ ⊢ A ∈ V | ||
Theorem | ssexg 3866 | 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 3867 | A subclass of a set is a set. Deduction form of ssexg 3866. (Contributed by David Moews, 1-May-2017.) |
⊢ (φ → B ∈ 𝐶) & ⊢ (φ → A ⊆ B) ⇒ ⊢ (φ → A ∈ V) | ||
Theorem | difexg 3868 | Existence of a difference. (Contributed by NM, 26-May-1998.) |
⊢ (A ∈ 𝑉 → (A ∖ B) ∈ V) | ||
Theorem | zfausab 3869* | Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.) |
⊢ A ∈ V ⇒ ⊢ {x ∣ (x ∈ A ∧ φ)} ∈ V | ||
Theorem | rabexg 3870* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) |
⊢ (A ∈ 𝑉 → {x ∈ A ∣ φ} ∈ V) | ||
Theorem | rabex 3871* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.) |
⊢ A ∈ V ⇒ ⊢ {x ∈ A ∣ φ} ∈ V | ||
Theorem | elssabg 3872* | Membership in a class abstraction involving a subset. Unlike elabg 2661, 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 3873* | The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
⊢ (∃x x ∈ A → ∩ A ∈ V) | ||
Theorem | intexr 3874 | If the intersection of a class exists, the class is non-empty. (Contributed by Jim Kingdon, 27-Aug-2018.) |
⊢ (∩ A ∈ V → A ≠ ∅) | ||
Theorem | intnexr 3875 | 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 3876 | The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
⊢ (∃xφ → ∩ {x ∣ φ} ∈ V) | ||
Theorem | intexrabim 3877 | 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 3878* | 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 3879* | 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 3880 | 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 3881 | 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 3882 | 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 3883 | No set equals its power set. The sethood antecedent is necessary; compare pwv 3549. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
⊢ (A ∈ 𝑉 → 𝒫 A ≠ A) | ||
Theorem | repizf2lem 3884 | Lemma for repizf2 3885. 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 3885* | Replacement. This version of replacement is stronger than repizf 3843 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 3843 with ax-sep 3845. 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 3886* | 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 3887 | Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.) |
⊢ ∅ ∈ 𝒫 A | ||
Theorem | 0nep0 3888 | The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.) |
⊢ ∅ ≠ {∅} | ||
Theorem | 0inp0 3889 | 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 3890 | The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) |
⊢ ∪ (A ∖ {∅}) = ∪ A | ||
Theorem | iin0imm 3891* | 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 3892* | If an indexed intersection of the empty set is empty, the index set is non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
⊢ (∩ x ∈ A ∅ = ∅ → A ≠ ∅) | ||
Theorem | intv 3893 | The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.) |
⊢ ∩ V = ∅ | ||
Theorem | axpweq 3894* | Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3897 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 3895* | A very strong generalization of the Axiom of Replacement (compare zfrep6 3844). 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 3842. (Contributed by NM, 17-Oct-2004.) |
⊢ (∀x ∈ z ∃yφ → ∃w∀x ∈ z ∃y ∈ w φ) | ||
Theorem | bnd2 3896* | A variant of the Boundedness Axiom bnd 3895 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 3897* |
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 3848).
The variant axpow2 3899 uses explicit subset notation. A version using class notation is pwex 3902. (Contributed by NM, 5-Aug-1993.) |
⊢ ∃y∀z(∀w(w ∈ z → w ∈ x) → z ∈ y) | ||
Theorem | zfpow 3898* | 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 3899* | A variant of the Axiom of Power Sets ax-pow 3897 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃y∀z(z ⊆ x → z ∈ y) | ||
Theorem | axpow3 3900* | A variant of the Axiom of Power Sets ax-pow 3897. 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) |
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