P. 43 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4201-4300   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-symdif 4201	Define the symmetric difference of two classes. Alternate definitions are dfsymdif2 4209, dfsymdif3 4254 and dfsymdif4 4207. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
Theorem	symdifcom 4202	Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012.)
⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴)
Theorem	symdifeq1 4203	Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
⊢ (𝐴 = 𝐵 → (𝐴 △ 𝐶) = (𝐵 △ 𝐶))
Theorem	symdifeq2 4204	Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
⊢ (𝐴 = 𝐵 → (𝐶 △ 𝐴) = (𝐶 △ 𝐵))
Theorem	nfsymdif 4205	Hypothesis builder for symmetric difference. (Contributed by Scott Fenton, 19-Feb-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 △ 𝐵)
Theorem	elsymdif 4206	Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
Theorem	dfsymdif4 4207*	Alternate definition of the symmetric difference. (Contributed by NM, 17-Aug-2004.) (Revised by AV, 17-Aug-2022.)
⊢ (𝐴 △ 𝐵) = {𝑥 ∣ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)}
Theorem	elsymdifxor 4208	Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020.) (Proof shortened by BJ, 13-Aug-2022.)
⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))
Theorem	dfsymdif2 4209*	Alternate definition of the symmetric difference. (Contributed by BJ, 30-Apr-2020.)
⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)}
Theorem	symdifass 4210	Symmetric difference is associative. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by BJ, 7-Sep-2022.)
⊢ ((𝐴 △ 𝐵) △ 𝐶) = (𝐴 △ (𝐵 △ 𝐶))
Theorem	difsssymdif 4211	The symmetric difference contains one of the differences. (Proposed by BJ, 18-Aug-2022.) (Contributed by AV, 19-Aug-2022.)
⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵)
Theorem	difsymssdifssd 4212	If the symmetric difference is contained in 𝐶, so is one of the differences. (Contributed by AV, 17-Aug-2022.)
⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)
2.1.13.5  Combinations of difference, union, and intersection of two classes
Theorem	unabs 4213	Absorption law for union. (Contributed by NM, 16-Apr-2006.)
⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴
Theorem	inabs 4214	Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴
Theorem	nssinpss 4215	Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴)
Theorem	nsspssun 4216	Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵))
Theorem	dfss4 4217	Subclass defined in terms of class difference. See comments under dfun2 4218. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴)
Theorem	dfun2 4218	An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 4219 and dfss4 4217 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation ∖ (class difference). (Contributed by NM, 10-Jun-2004.)
⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
Theorem	dfin2 4219	An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4218. Another version is given by dfin4 4226. (Contributed by NM, 10-Jun-2004.)
⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵))
Theorem	difin 4220	Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵)
Theorem	ssdifim 4221	Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022.)
⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵))
Theorem	ssdifsym 4222	Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.)
⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) ↔ 𝐴 = (𝑉 ∖ 𝐵)))
Theorem	dfss5 4223*	Alternate definition of subclass relationship: a class 𝐴 is a subclass of another class 𝐵 iff each element of 𝐴 is equal to an element of 𝐵. (Contributed by AV, 13-Nov-2020.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝑦)
Theorem	dfun3 4224	Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
Theorem	dfin3 4225	Intersection defined in terms of union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
Theorem	dfin4 4226	Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)
⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
Theorem	invdif 4227	Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)
Theorem	indif 4228	Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)
Theorem	indif2 4229	Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
Theorem	indif1 4230	Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
Theorem	indifcom 4231	Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶))
Theorem	indi 4232	Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶))
Theorem	undi 4233	Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ∪ (𝐵 ∩ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶))
Theorem	indir 4234	Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶))
Theorem	undir 4235	Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶))
Theorem	unineq 4236	Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)
⊢ (((𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) ↔ 𝐴 = 𝐵)
Theorem	uneqin 4237	Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) ↔ 𝐴 = 𝐵)
Theorem	difundi 4238	Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))
Theorem	difundir 4239	Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))
Theorem	difindi 4240	Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))
Theorem	difindir 4241	Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (𝐵 ∖ 𝐶))
Theorem	indifdi 4242	Distribute intersection over difference. (Contributed by BTernaryTau, 14-Aug-2024.)
⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶))
Theorem	indifdir 4243	Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by BTernaryTau, 14-Aug-2024.)
⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶))
Theorem	difdif2 4244	Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)
⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶))
Theorem	undm 4245	De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
⊢ (V ∖ (𝐴 ∪ 𝐵)) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵))
Theorem	indm 4246	De Morgan's law for intersection. Theorem 5.2(13') of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
⊢ (V ∖ (𝐴 ∩ 𝐵)) = ((V ∖ 𝐴) ∪ (V ∖ 𝐵))
Theorem	difun1 4247	A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶)
Theorem	undif3 4248	An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.) (Proof shortened by JJ, 13-Jul-2021.)
⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))
Theorem	difin2 4249	Represent a class difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = ((𝐶 ∖ 𝐵) ∩ 𝐴))
Theorem	dif32 4250	Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
Theorem	difabs 4251	Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
Theorem	sscon34b 4252	Relative complementation reverses inclusion of subclasses. Relativized version of complss 4099. (Contributed by RP, 3-Jun-2021.)
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)))
Theorem	rcompleq 4253	Two subclasses are equal if and only if their relative complements are equal. Relativized version of compleq 4100. (Contributed by RP, 10-Jun-2021.)
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 = 𝐵 ↔ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)))
Theorem	dfsymdif3 4254	Alternate definition of the symmetric difference, given in Example 4.1 of [Stoll] p. 262 (the original definition corresponds to [Stoll] p. 13). (Contributed by NM, 17-Aug-2004.) (Revised by BJ, 30-Apr-2020.)
⊢ (𝐴 △ 𝐵) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))
2.1.13.6  Class abstractions with difference, union, and intersection of two classes
Theorem	unabw 4255*	Union of two class abstractions. Version of unab 4256 using implicit substitution, which does not require ax-8 2112, ax-10 2143, ax-12 2179. (Contributed by GG, 15-Oct-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    ⇒   ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑦 ∣ (𝜒 ∨ 𝜃)}
Theorem	unab 4256	Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∨ 𝜓)}
Theorem	inab 4257	Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ({𝑥 ∣ 𝜑} ∩ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ 𝜓)}
Theorem	difab 4258	Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ({𝑥 ∣ 𝜑} ∖ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)}
Theorem	abanssl 4259	A class abstraction with a conjunction is a subset of the class abstraction with the left conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
⊢ {𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜑}
Theorem	abanssr 4260	A class abstraction with a conjunction is a subset of the class abstraction with the right conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
⊢ {𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜓}
Theorem	notabw 4261*	A class abstraction defined by a negation. Version of notab 4262 using implicit substitution, which does not require ax-10 2143, ax-12 2179. (Contributed by GG, 15-Oct-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜓})
Theorem	notab 4262	A class abstraction defined by a negation. (Contributed by FL, 18-Sep-2010.)
⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑})
Theorem	unrab 4263	Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)}
Theorem	inrab 4264	Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)}
Theorem	inrab2 4265*	Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ 𝜑}
Theorem	difrab 4266	Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}
Theorem	dfrab3 4267*	Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})
Theorem	dfrab2 4268*	Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴)
Theorem	rabdif 4269*	Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑}
Theorem	notrab 4270*	Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
Theorem	dfrab3ss 4271*	Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)
⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}))
Theorem	rabun2 4272	Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜑} = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜑})
2.1.13.7  Restricted uniqueness with difference, union, and intersection
Theorem	reuun2 4273	Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Wolf Lammen, 15-May-2025.)
⊢ (¬ ∃𝑥 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜑))
Theorem	reuss2 4274*	Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑)
Theorem	reuss 4275*	Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑)
Theorem	reuun1 4276*	Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑)
Theorem	reupick 4277*	Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	reupick3 4278*	Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓))
Theorem	reupick2 4279*	Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))
Theorem	euelss 4280*	Transfer uniqueness of an element to a smaller subclass. (Contributed by AV, 14-Apr-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴 ∧ ∃!𝑥 𝑥 ∈ 𝐵) → ∃!𝑥 𝑥 ∈ 𝐴)
2.1.14  The empty set
Syntax	c0 4281	Extend class notation to include the empty set.
class ∅
Definition	df-nul 4282	Define the empty set. More precisely, we should write "empty class". It will be posited in ax-nul 5242 that an empty set exists. Then, by uniqueness among classes (eq0 4298, as opposed to the weaker uniqueness among sets, nulmo 2707), it will follow that ∅ is indeed a set (0ex 5243). Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 4284. (Contributed by NM, 17-Jun-1993.) Clarify that at this point, it is not established that it is a set. (Revised by BJ, 22-Sep-2022.)
⊢ ∅ = (V ∖ V)
Theorem	dfnul4 4283	Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2112, df-clel 2804. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4284. (Revised by BJ, 23-Sep-2024.)
⊢ ∅ = {𝑥 ∣ ⊥}
Theorem	dfnul2 4284	Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2143, ax-11 2159, and ax-12 2179. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Theorem	dfnul3 4285	Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)
⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}
Theorem	noel 4286	The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) Remove dependency on ax-10 2143, ax-11 2159, and ax-12 2179. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
⊢ ¬ 𝐴 ∈ ∅
Theorem	nel02 4287	The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.)
⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)
Theorem	n0i 4288	If a class has elements, then it is not empty. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)
Theorem	ne0i 4289	If a class has elements, then it is nonempty. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)
Theorem	ne0d 4290	Deduction form of ne0i 4289. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ≠ ∅)
Theorem	n0ii 4291	If a class has elements, then it is not empty. Inference associated with n0i 4288. (Contributed by BJ, 15-Jul-2021.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ¬ 𝐵 = ∅
Theorem	ne0ii 4292	If a class has elements, then it is nonempty. Inference associated with ne0i 4289. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐵 ≠ ∅
Theorem	vn0 4293	The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2112, df-clel 2804. (Revised by GG, 6-Sep-2024.)
⊢ V ≠ ∅
Theorem	vn0ALT 4294	Alternate proof of vn0 4293. Shorter, but requiring df-clel 2804, ax-8 2112. (Contributed by NM, 11-Sep-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ V ≠ ∅
Theorem	eq0f 4295	A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
Theorem	neq0f 4296	A class is not empty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of neq0 4300 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
Theorem	n0f 4297	A class is nonempty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 4301 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by NM, 17-Oct-2003.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
Theorem	eq0 4298*	A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2159, ax-12 2179. (Revised by GG and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2112, df-clel 2804. (Revised by GG, 6-Sep-2024.)
⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
Theorem	eq0ALT 4299*	Alternate proof of eq0 4298. Shorter, but requiring df-clel 2804, ax-8 2112. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2159, ax-12 2179. (Revised by GG and Steven Nguyen, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
Theorem	neq0 4300*	A class is not empty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 21-Jun-1993.) Avoid ax-11 2159, ax-12 2179. (Revised by GG, 28-Jun-2024.)
⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)