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	axsep2 4101*	A less restrictive version of the Separation Scheme ax-sep 4100, 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 4100 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 4102*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4100, we invoke the Axiom of Extensionality (indirectly via vtocl 2780), 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 4103*	Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4100. 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 4104*	Derive a weakened version of ax-i9 1518, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4100 and Extensionality ax-ext 2147. The theorem ¬ ∀𝑥¬ 𝑥 = 𝑦 also holds (ax9vsep 4105), 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 4105*	Derive a weakened version of ax-9 1519, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4100 and Extensionality ax-ext 2147. In intuitionistic logic a9evsep 4104 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 4106*	Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2150 to strengthen the hypothesis in the form of axnul 4107). (Contributed by NM, 22-Dec-2007.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥    ⇒   ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	axnul 4107*	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 4100. 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 4106). This theorem should not be referenced by any proof. Instead, use ax-nul 4108 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 4108*	The Null Set Axiom of IZF set theory. It was derived as axnul 4107 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 4109	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 4108. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ∅ ∈ V
Theorem	csbexga 4110	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 4111	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 4112*	No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	vnex 4113	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	vprc 4114	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 4115	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
⊢ ¬ V ∈ 𝐴
Theorem	inex1 4116	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 4117	Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	inex1g 4118	Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	ssex 4119	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 4100 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	ssexi 4120	The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	ssexg 4121	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 4122	A subclass of a set is a set. Deduction form of ssexg 4121. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	difexg 4123	Existence of a difference. (Contributed by NM, 26-May-1998.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	zfausab 4124*	Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V
Theorem	rabexg 4125*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	rabex 4126*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
Theorem	elssabg 4127*	Membership in a class abstraction involving a subset. Unlike elabg 2872, 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 4128*	The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
Theorem	intexr 4129	If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)
Theorem	intnexr 4130	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 4131	The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∃𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} ∈ V)
Theorem	intexrabim 4132	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 4133*	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 4134*	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 4135	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 4136	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 4137	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 4138	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 4139	No set equals its power set. The sethood antecedent is necessary; compare pwv 3788. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)
Theorem	repizf2lem 4140	Lemma for repizf2 4141. 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 4141*	Replacement. This version of replacement is stronger than repizf 4098 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 4098 with ax-sep 4100. 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 4142*	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 4143	Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
⊢ ∅ ∈ 𝒫 𝐴
Theorem	0nep0 4144	The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
⊢ ∅ ≠ {∅}
Theorem	0inp0 4145	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 4146	The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴
Theorem	iin0imm 4147*	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 4148*	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 4149	The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
⊢ ∩ V = ∅
Theorem	axpweq 4150*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4153 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 4151*	A very strong generalization of the Axiom of Replacement (compare zfrep6 4099). 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 4097. (Contributed by NM, 17-Oct-2004.)
⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
Theorem	bnd2 4152*	A variant of the Boundedness Axiom bnd 4151 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 4153*	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 4103). The variant axpow2 4155 uses explicit subset notation. A version using class notation is pwex 4162. (Contributed by NM, 5-Aug-1993.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	zfpow 4154*	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 4155*	A variant of the Axiom of Power Sets ax-pow 4153 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
Theorem	axpow3 4156*	A variant of the Axiom of Power Sets ax-pow 4153. 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 4157*	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 4158	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 4159 from vpwex 4158. (Revised by BJ, 10-Aug-2022.)
⊢ 𝒫 𝑥 ∈ V
Theorem	pwexg 4159	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 4160	Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
Theorem	abssexg 4161*	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 4162	Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝒫 𝐴 ∈ V
Theorem	snexg 4163	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 4164	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 4165	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 4166	A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
⊢ ¬ ¬ {𝐴} ∈ V
Theorem	p0ex 4167	The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
⊢ {∅} ∈ V
Theorem	pp0ex 4168	{∅, {∅}} (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
⊢ {∅, {∅}} ∈ V
Theorem	ord3ex 4169	The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
⊢ {∅, {∅}, {∅, {∅}}} ∈ V
Theorem	dtruarb 4170*	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 4536 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 4171	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 4172*	Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3489 and undifdcss 6888 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 4173	Formula for an abbreviation of excluded middle.
wff EXMID
Definition	df-exmid 4174	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 4172 with exmidundif 4185. 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 4175 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 4100, in which case EXMID means that all propositions are decidable (see exmidexmid 4175 and notice that it relies on ax-sep 4100). If we instead work with ax-bdsep 13766, 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 4175	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 833, peircedc 904, or condc 843. (Contributed by Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID → DECID 𝜑)
Theorem	ss1o0el1 4176	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 4177	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 = {∅})))
Theorem	pwntru 4178	A slight strengthening of pwtrufal 13877. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
⊢ ((𝐴 ⊆ {∅} ∧ 𝐴 ≠ {∅}) → 𝐴 = ∅)
Theorem	exmid1dc 4179*	A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4172 or ordtriexmid 4498. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})    ⇒   ⊢ (𝜑 → EXMID)
Theorem	exmidn0m 4180*	Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
⊢ (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
Theorem	exmidsssn 4181*	Excluded middle is equivalent to the biconditionalized version of sssnr 3733 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
Theorem	exmidsssnc 4182*	Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4177 but lets you choose any set as the element of the singleton rather than just ∅. It is similar to exmidsssn 4181 but for a particular set 𝐵 rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
⊢ (𝐵 ∈ 𝑉 → (EXMID ↔ ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
Theorem	exmid0el 4183	Excluded middle is equivalent to decidability of ∅ being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥)
Theorem	exmidel 4184*	Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦)
Theorem	exmidundif 4185*	Excluded middle is equivalent to every subset having a complement. That is, the union of a subset and its relative complement being the whole set. Although special cases such as undifss 3489 and undifdcss 6888 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
Theorem	exmidundifim 4186*	Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4185 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
2.3.3  Axiom of Pairing
Axiom	ax-pr 4187*	The Axiom of Pairing of IZF set theory. Axiom 2 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 4103). (Contributed by NM, 14-Nov-2006.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	zfpair2 4188	Derive the abbreviated version of the Axiom of Pairing from ax-pr 4187. (Contributed by NM, 14-Nov-2006.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	prexg 4189	The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3686, prprc1 3684, and prprc2 3685. (Contributed by Jim Kingdon, 16-Sep-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
Theorem	snelpwi 4190	A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)
Theorem	snelpw 4191	A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
Theorem	prelpwi 4192	A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)
Theorem	rext 4193*	A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)
Theorem	sspwb 4194	Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Theorem	unipw 4195	A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
⊢ ∪ 𝒫 𝐴 = 𝐴
Theorem	pwel 4196	Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)
Theorem	pwtr 4197	A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴)
Theorem	ssextss 4198*	An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
Theorem	ssext 4199*	An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
Theorem	nssssr 4200*	Negation of subclass relationship. Compare nssr 3202. (Contributed by Jim Kingdon, 17-Sep-2018.)
⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵)