P. 40 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	invdif 3901	Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)
Theorem	indif 3902	Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)
Theorem	indif2 3903	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 3904	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 3905	Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶))
Theorem	indi 3906	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 3907	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 3908	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 3909	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 3910	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 3911	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 3912	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 3913	Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))
Theorem	difindi 3914	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 3915	Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (𝐵 ∖ 𝐶))
Theorem	indifdir 3916	Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶))
Theorem	difdif2 3917	Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)
⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶))
Theorem	undm 3918	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 3919	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 3920	A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶)
Theorem	undif3 3921	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	undif3OLD 3922	Obsolete proof of undif3 3921 as of 13-Jul-2021. (Contributed by Alan Sare, 17-Apr-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))
Theorem	difin2 3923	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 3924	Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵)
Theorem	difabs 3925	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	dfsymdif3 3926	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	unab 3927	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 3928	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 3929	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	notab 3930	A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑})
Theorem	unrab 3931	Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)}
Theorem	inrab 3932	Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)}
Theorem	inrab2 3933*	Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ 𝜑}
Theorem	difrab 3934	Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}
Theorem	dfrab3 3935*	Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})
Theorem	dfrab2 3936*	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	notrab 3937*	Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
Theorem	dfrab3ss 3938*	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 3939	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	reuss2 3940*	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 3941*	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 3942*	Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑)
Theorem	reuun2 3943*	Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
⊢ (¬ ∃𝑥 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜑))
Theorem	reupick 3944*	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 3945*	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 3946*	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 3947*	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 3948	Extend class notation to include the empty set.
class ∅
Definition	df-nul 3949	Define the empty set. Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 3950. (Contributed by NM, 17-Jun-1993.)
⊢ ∅ = (V ∖ V)
Theorem	dfnul2 3950	Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Theorem	dfnul3 3951	Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}
Theorem	noel 3952	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.)
⊢ ¬ 𝐴 ∈ ∅
Theorem	n0i 3953	If a set has elements, then it is not empty. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅)
Theorem	ne0i 3954	If a set has elements, then it is not empty. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅)
Theorem	n0ii 3955	If a set has elements, then it is not empty. Inference associated with n0i 3953. (Contributed by BJ, 15-Jul-2021.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ¬ 𝐵 = ∅
Theorem	ne0ii 3956	If a set has elements, then it is not empty. Inference associated with ne0i 3954. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐵 ≠ ∅
Theorem	vn0 3957	The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
⊢ V ≠ ∅
Theorem	eq0f 3958	The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
Theorem	neq0f 3959	A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of neq0 3963 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 3960	A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3964 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	n0fOLD 3961	Obsolete proof of n0f 3960 as of 15-Jul-2021. (Contributed by NM, 17-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
Theorem	eq0 3962*	The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
Theorem	neq0 3963*	A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 21-Jun-1993.)
⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
Theorem	n0 3964*	A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 29-Sep-2006.)
⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
Theorem	nel0 3965*	From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
⊢ ¬ 𝑥 ∈ 𝐴    ⇒   ⊢ 𝐴 = ∅
Theorem	reximdva0 3966*	Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	rspn0 3967*	Specialization for restricted generalization with a nonempty set. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
Theorem	n0rex 3968*	There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.)
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴)
Theorem	ssn0rex 3969*	There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴)
Theorem	n0moeu 3970*	A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
⊢ (𝐴 ≠ ∅ → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴))
Theorem	rex0 3971	Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
Theorem	0el 3972*	Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	n0el 3973*	Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥)
Theorem	eqeuel 3974*	A condition which implies the existence of a unique element of a class. (Contributed by AV, 4-Jan-2022.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥 ∈ 𝐴)
Theorem	ssdif0 3975	Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
Theorem	difn0 3976	If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)
⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵)
Theorem	pssdifn0 3977	A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅)
Theorem	pssdif 3978	A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)
Theorem	difin0ss 3979	Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
⊢ (((𝐴 ∖ 𝐵) ∩ 𝐶) = ∅ → (𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵))
Theorem	inssdif0 3980	Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)
Theorem	difid 3981	The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)
⊢ (𝐴 ∖ 𝐴) = ∅
Theorem	difidALT 3982	Alternate proof of difid 3981. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∖ 𝐴) = ∅
Theorem	dif0 3983	The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∖ ∅) = 𝐴
Theorem	ab0 3984	The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 3987 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne 2824). (Contributed by BJ, 19-Mar-2021.)
⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
Theorem	dfnf5 3985	Characterization of non-freeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V))
Theorem	ab0orv 3986*	The class builder of a wff not containing the abstraction variable is either the empty set or the universal class. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.)
⊢ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V)
Theorem	abn0 3987	Nonempty class abstraction. See also ab0 3984. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
Theorem	rab0 3988	Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Theorem	rab0OLD 3989	Obsolete proof of rab0 3988 as of 14-Jul-2021. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Theorem	rabeq0 3990	Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.) (Revised by BJ, 16-Jul-2021.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
Theorem	rabn0 3991	Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑)
Theorem	rabn0OLD 3992	Obsolete proof of rabn0 3991 as of 16-Jul-2021. (Contributed by NM, 29-Aug-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑)
Theorem	rabeq0OLD 3993	Obsolete proof of rabeq0 3990 as of 16-Jul-2021. (Contributed by Jeff Madsen, 7-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
Theorem	rabxm 3994*	Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
Theorem	rabnc 3995*	Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅
Theorem	elneldisj 3996*	The set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element are disjoint. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}    &   ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶}    ⇒   ⊢ (𝐸 ∩ 𝑁) = ∅
Theorem	elnelun 3997*	The union of the set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element yields the original set. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}    &   ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶}    ⇒   ⊢ (𝐸 ∪ 𝑁) = 𝐴
Theorem	elneldisjOLD 3998*	Obsolete version of elneldisj 3996 as of 17-Dec-2021. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝑠}    &   ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝑠}    ⇒   ⊢ (𝐸 ∩ 𝑁) = ∅
Theorem	elnelunOLD 3999*	Obsolete version of elnelun 3997 as of 17-Dec-2021. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝑠}    &   ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝑠}    ⇒   ⊢ (𝐸 ∪ 𝑁) = 𝐴
Theorem	un0 4000	The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 15-Jul-1993.)
⊢ (𝐴 ∪ ∅) = 𝐴