P. 42 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cbvmptv 4101*	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 4102*	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 4103	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴
Definition	df-tr 4104	Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4105 (which is suggestive of the word "transitive"), dftr3 4107, dftr4 4108, and dftr5 4106. 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 4105*	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 4106*	An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
Theorem	dftr3 4107*	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 4108	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 4109	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Theorem	trel 4110	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 4111	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 4112	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 4113	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
Theorem	tr0 4114	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
⊢ Tr ∅
Theorem	trv 4115	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
⊢ Tr V
Theorem	triun 4116*	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 4117*	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 4118*	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 4119*	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 4120*	Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4174 but uses a freeness hypothesis in place of one of the distinct variable conditions. (Contributed by Jim Kingdon, 23-Aug-2018.)
⊢ Ⅎ𝑏𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑)
Theorem	repizf 4121*	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 4120. It is identical to zfrep6 4122 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.)
⊢ Ⅎ𝑏𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝑎 ∃!𝑦𝜑 → ∃𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑)
Theorem	zfrep6 4122*	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 4123 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 4123*	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 disjoint variable condition 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 2963. 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 4124*	A less restrictive version of the Separation Scheme ax-sep 4123, 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 4123 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 4125*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4123, we invoke the Axiom of Extensionality (indirectly via vtocl 2793), 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 4126*	Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4123. 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 4127*	Derive a weakened version of ax-i9 1530, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4123 and Extensionality ax-ext 2159. The theorem ¬ ∀𝑥¬ 𝑥 = 𝑦 also holds (ax9vsep 4128), 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 4128*	Derive a weakened version of ax-9 1531, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4123 and Extensionality ax-ext 2159. In intuitionistic logic a9evsep 4127 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 4129*	Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2162 to strengthen the hypothesis in the form of axnul 4130). (Contributed by NM, 22-Dec-2007.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥    ⇒   ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	axnul 4130*	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 4123. 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 4129). This theorem should not be referenced by any proof. Instead, use ax-nul 4131 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 4131*	The Null Set Axiom of IZF set theory. It was derived as axnul 4130 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 4132	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 4131. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ∅ ∈ V
Theorem	csbexga 4133	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 4134	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 4135*	No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	vnex 4136	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	vprc 4137	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 4138	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
⊢ ¬ V ∈ 𝐴
Theorem	inex1 4139	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 4140	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	inex1g 4141	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	ssex 4142	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 4123 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	ssexi 4143	The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	ssexg 4144	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 4145	A subclass of a set is a set. Deduction form of ssexg 4144. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	difexg 4146	Existence of a difference. (Contributed by NM, 26-May-1998.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	zfausab 4147*	Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V
Theorem	rabexg 4148*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	rabex 4149*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
Theorem	elssabg 4150*	Membership in a class abstraction involving a subset. Unlike elabg 2885, 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 4151*	The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
Theorem	intexr 4152	If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)
Theorem	intnexr 4153	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 4154	The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∃𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} ∈ V)
Theorem	intexrabim 4155	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 4156*	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 4157*	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 4158	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 4159	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	elpwi2 4160	Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
⊢ 𝐵 ∈ 𝑉    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ 𝒫 𝐵
Theorem	pwnss 4161	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 4162	No set equals its power set. The sethood antecedent is necessary; compare pwv 3810. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)
Theorem	repizf2lem 4163	Lemma for repizf2 4164. 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 4164*	Replacement. This version of replacement is stronger than repizf 4121 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 4121 with ax-sep 4123. 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 4165*	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 4166	Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
⊢ ∅ ∈ 𝒫 𝐴
Theorem	0nep0 4167	The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
⊢ ∅ ≠ {∅}
Theorem	0inp0 4168	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 4169	The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴
Theorem	iin0imm 4170*	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 4171*	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 4172	The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
⊢ ∩ V = ∅
Theorem	axpweq 4173*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4176 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 4174*	A very strong generalization of the Axiom of Replacement (compare zfrep6 4122). 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 4120. (Contributed by NM, 17-Oct-2004.)
⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
Theorem	bnd2 4175*	A variant of the Boundedness Axiom bnd 4174 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 4176*	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 4126). The variant axpow2 4178 uses explicit subset notation. A version using class notation is pwex 4185. (Contributed by NM, 5-Aug-1993.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	zfpow 4177*	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 4178*	A variant of the Axiom of Power Sets ax-pow 4176 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
Theorem	axpow3 4179*	A variant of the Axiom of Power Sets ax-pow 4176. 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 4180*	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 4181	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 4182 from vpwex 4181. (Revised by BJ, 10-Aug-2022.)
⊢ 𝒫 𝑥 ∈ V
Theorem	pwexg 4182	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	pwexd 4183	Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
Theorem	abssexg 4184*	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 4185	Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝒫 𝐴 ∈ V
Theorem	snexg 4186	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 4187	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 4188	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 4189	A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
⊢ ¬ ¬ {𝐴} ∈ V
Theorem	p0ex 4190	The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
⊢ {∅} ∈ V
Theorem	pp0ex 4191	{∅, {∅}} (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
⊢ {∅, {∅}} ∈ V
Theorem	ord3ex 4192	The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
⊢ {∅, {∅}, {∅, {∅}}} ∈ V
Theorem	dtruarb 4193*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4560 in which we are given a set 𝑦 and go from there to a set 𝑥 which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦
Theorem	pwuni 4194	A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
Theorem	undifexmid 4195*	Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3505 and undifdcss 6925 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
2.3.2  A notation for excluded middle
Syntax	wem 4196	Formula for an abbreviation of excluded middle.
wff EXMID
Definition	df-exmid 4197	The expression EXMID will be used as a readable shorthand for any form of the law of the excluded middle; this is a useful shorthand largely because it hides statements of the form "for any proposition" in a system which can only quantify over sets, not propositions. To see how this compares with other ways of expressing excluded middle, compare undifexmid 4195 with exmidundif 4208. The former may be more recognizable as excluded middle because it is in terms of propositions, and the proof may be easier to follow for much the same reason (it just has to show 𝜑 and ¬ 𝜑 in the the relevant parts of the proof). The latter, however, has the key advantage of being able to prove both directions of the biconditional. To state that excluded middle implies a proposition is hard to do gracefully without EXMID, because there is no way to write a hypothesis 𝜑 ∨ ¬ 𝜑 for an arbitrary proposition; instead the hypothesis would need to be the particular instance of excluded middle which that proof needs. Or to say it another way, EXMID implies DECID 𝜑 by exmidexmid 4198 but there is no good way to express the converse. This definition and how we use it is easiest to understand (and most appropriate to assign the name "excluded middle" to) if we assume ax-sep 4123, in which case EXMID means that all propositions are decidable (see exmidexmid 4198 and notice that it relies on ax-sep 4123). If we instead work with ax-bdsep 14824, EXMID as defined here means that all bounded propositions are decidable. (Contributed by Mario Carneiro and Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
Theorem	exmidexmid 4198	EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions. To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 843, peircedc 914, or condc 853. (Contributed by Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID → DECID 𝜑)
Theorem	ss1o0el1 4199	A subclass of {∅} contains the empty set if and only if it equals {∅}. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)
⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅}))
Theorem	exmid01 4200	Excluded middle is equivalent to saying any subset of {∅} is either ∅ or {∅}. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))