P. 41 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cbvopab1 4001*	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.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Theorem	cbvopab2 4002*	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 4003*	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 4004*	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 4005*	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 4006*	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 4007	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 4008	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 4009*	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 4010*	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 4011*	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 4012*	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 4013*	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 4014	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 4015	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 4016	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 4017	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 4018*	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 4019*	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 4020*	Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵)
Theorem	nfmpt1 4021	Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	cbvmptf 4022*	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 4023*	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 4024*	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 4025*	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
Syntax	wtr 4026	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴
Definition	df-tr 4027	Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4028 (which is suggestive of the word "transitive"), dftr3 4030, dftr4 4031, and dftr5 4029. 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 4028*	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 4029*	An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
Theorem	dftr3 4030*	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 4031	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 4032	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Theorem	trel 4033	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 4034	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 4035	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 4036	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
Theorem	tr0 4037	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
⊢ Tr ∅
Theorem	trv 4038	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
⊢ Tr V
Theorem	triun 4039*	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 4040*	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 4041*	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 4042*	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)
2.2  IZF Set Theory - add the Axioms of Collection and Separation
2.2.1  Introduce the Axiom of Collection
Axiom	ax-coll 4043*	Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4096 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 4044*	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 4043. It is identical to zfrep6 4045 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 4045*	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 4046 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)
2.2.2  Introduce the Axiom of Separation
Axiom	ax-sep 4046*	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 2908. 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 4047*	A less restrictive version of the Separation Scheme ax-sep 4046, 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 4046 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 4048*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4046, we invoke the Axiom of Extensionality (indirectly via vtocl 2740), 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 4049*	Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4046. 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 4050*	Derive a weakened version of ax-i9 1510, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4046 and Extensionality ax-ext 2121. The theorem ¬ ∀𝑥¬ 𝑥 = 𝑦 also holds (ax9vsep 4051), 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 4051*	Derive a weakened version of ax-9 1511, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4046 and Extensionality ax-ext 2121. In intuitionistic logic a9evsep 4050 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
Theorem	zfnuleu 4052*	Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2124 to strengthen the hypothesis in the form of axnul 4053). (Contributed by NM, 22-Dec-2007.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥    ⇒   ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	axnul 4053*	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 4046. 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 4052). This theorem should not be referenced by any proof. Instead, use ax-nul 4054 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 4054*	The Null Set Axiom of IZF set theory. It was derived as axnul 4053 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 4055	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 4054. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ∅ ∈ V
Theorem	csbexga 4056	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 4057	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
2.2.4  Theorems requiring subset and intersection existence
Theorem	nalset 4058*	No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	vnex 4059	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	vprc 4060	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 4061	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
⊢ ¬ V ∈ 𝐴
Theorem	inex1 4062	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 4063	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	inex1g 4064	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	ssex 4065	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 4046 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	ssexi 4066	The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	ssexg 4067	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 4068	A subclass of a set is a set. Deduction form of ssexg 4067. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	difexg 4069	Existence of a difference. (Contributed by NM, 26-May-1998.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	zfausab 4070*	Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V
Theorem	rabexg 4071*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	rabex 4072*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
Theorem	elssabg 4073*	Membership in a class abstraction involving a subset. Unlike elabg 2830, 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 4074*	The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
Theorem	intexr 4075	If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)
Theorem	intnexr 4076	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 4077	The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∃𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} ∈ V)
Theorem	intexrabim 4078	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 4079*	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 4080*	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 4081	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 4082	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 4083	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 4084	No set equals its power set. The sethood antecedent is necessary; compare pwv 3735. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)
Theorem	repizf2lem 4085	Lemma for repizf2 4086. 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 4086*	Replacement. This version of replacement is stronger than repizf 4044 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 4044 with ax-sep 4046. 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)
2.2.5  Theorems requiring empty set existence
Theorem	class2seteq 4087*	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 4088	Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
⊢ ∅ ∈ 𝒫 𝐴
Theorem	0nep0 4089	The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
⊢ ∅ ≠ {∅}
Theorem	0inp0 4090	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 4091	The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴
Theorem	iin0imm 4092*	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 4093*	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 4094	The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
⊢ ∩ V = ∅
Theorem	axpweq 4095*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4098 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))
2.2.6  Collection principle
Theorem	bnd 4096*	A very strong generalization of the Axiom of Replacement (compare zfrep6 4045). 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 4043. (Contributed by NM, 17-Oct-2004.)
⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
Theorem	bnd2 4097*	A variant of the Boundedness Axiom bnd 4096 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))
2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
2.3.1  Introduce the Axiom of Power Sets
Axiom	ax-pow 4098*	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 4049). The variant axpow2 4100 uses explicit subset notation. A version using class notation is pwex 4107. (Contributed by NM, 5-Aug-1993.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	zfpow 4099*	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 4100*	A variant of the Axiom of Power Sets ax-pow 4098 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)