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Theorem List for Metamath Proof Explorer - 3801-3900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuneq12d 3801 Equality deduction for the union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremnfun 3802 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremunass 3803 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))

Theoremun12 3804 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))

Theoremun23 3805 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)

Theoremun4 3806 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))

Theoremunundi 3807 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremunundir 3808 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremssun1 3809 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
𝐴 ⊆ (𝐴𝐵)

Theoremssun2 3810 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
𝐴 ⊆ (𝐵𝐴)

Theoremssun3 3811 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ⊆ (𝐵𝐶))

Theoremssun4 3812 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
(𝐴𝐵𝐴 ⊆ (𝐶𝐵))

Theoremelun1 3813 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐵𝐶))

Theoremelun2 3814 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐶𝐵))

Theoremunss1 3815 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssequn1 3816 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)

Theoremunss2 3817 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremunss12 3818 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremssequn2 3819 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)

Theoremunss 3820 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)

Theoremunssi 3821 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
𝐴𝐶    &   𝐵𝐶       (𝐴𝐵) ⊆ 𝐶

Theoremunssd 3822 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Theoremunssad 3823 If (𝐴𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 3820. Partial converse of unssd 3822. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐴𝐶)

Theoremunssbd 3824 If (𝐴𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3820. Partial converse of unssd 3822. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐵𝐶)

Theoremssun 3825 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))

Theoremrexun 3826 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
(∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑))

Theoremralunb 3827 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑))

Theoremralun 3828 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑) → ∀𝑥 ∈ (𝐴𝐵)𝜑)

2.1.13.3  The intersection of two classes

Theoremelin 3829 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Theoremelini 3830 Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∈ (𝐵𝐶)

Theoremelind 3831 Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝑋𝐴)    &   (𝜑𝑋𝐵)       (𝜑𝑋 ∈ (𝐴𝐵))

Theoremelinel1 3832 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)

Theoremelinel2 3833 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐶)

Theoremelin2 3834 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = (𝐵𝐶)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶))

Theoremelin1d 3835 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
(𝜑𝑋 ∈ (𝐴𝐵))       (𝜑𝑋𝐴)

Theoremelin2d 3836 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
(𝜑𝑋 ∈ (𝐴𝐵))       (𝜑𝑋𝐵)

Theoremelin3 3837 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = ((𝐵𝐶) ∩ 𝐷)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Theoremincom 3838 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 21-Jun-1993.)
(𝐴𝐵) = (𝐵𝐴)

Theoremineqri 3839* Inference from membership to intersection. (Contributed by NM, 21-Jun-1993.)
((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶

Theoremineq1 3840 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremineq2 3841 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremineq12 3842 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremineq1i 3843 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremineq2i 3844 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremineq12i 3845 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremineq1d 3846 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremineq2d 3847 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremineq12d 3848 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremineqan12d 3849 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))

Theoremsseqin2 3850 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Theoremdfss1OLD 3851 Obsolete as of 22-Jul-2021. (Contributed by NM, 10-Jan-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Theoremdfss5OLD 3852 Obsolete as of 22-Jul-2021. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐵𝐴 = (𝐵𝐴))

Theoremnfin 3853 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremrabbi2dva 3854* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
((𝜑𝑥𝐴) → (𝑥𝐵𝜓))       (𝜑 → (𝐴𝐵) = {𝑥𝐴𝜓})

Theoreminidm 3855 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐴) = 𝐴

Theoreminass 3856 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))

Theoremin12 3857 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
(𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))

Theoremin32 3858 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)

Theoremin13 3859 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
(𝐴 ∩ (𝐵𝐶)) = (𝐶 ∩ (𝐵𝐴))

Theoremin31 3860 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)

Theoreminrot 3861 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐴) ∩ 𝐵)

Theoremin4 3862 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))

Theoreminindi 3863 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoreminindir 3864 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremsseqin2OLD 3865 Obsolete proof of sseqin2 3850 as of 22-Jul-2021. (Contributed by NM, 17-May-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Theoreminss1 3866 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
(𝐴𝐵) ⊆ 𝐴

Theoreminss2 3867 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
(𝐴𝐵) ⊆ 𝐵

Theoremssin 3868 Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))

Theoremssini 3869 An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
𝐴𝐵    &   𝐴𝐶       𝐴 ⊆ (𝐵𝐶)

Theoremssind 3870 A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐴 ⊆ (𝐵𝐶))

Theoremssrin 3871 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremsslin 3872 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremss2in 3873 Intersection of subclasses. (Contributed by NM, 5-May-2000.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremssinss1 3874 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
(𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)

Theoreminss 3875 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)

2.1.13.4  The symmetric difference of two classes

Syntaxcsymdif 3876 Declare the syntax for symmetric difference.
class (𝐴𝐵)

Definitiondf-symdif 3877 Define the symmetric difference of two classes. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) = ((𝐴𝐵) ∪ (𝐵𝐴))

Theoremsymdifcom 3878 Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴𝐵) = (𝐵𝐴)

Theoremsymdifeq1 3879 Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremsymdifeq2 3880 Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremnfsymdif 3881 Hypothesis builder for symmetric difference. (Contributed by Scott Fenton, 19-Feb-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremelsymdif 3882 Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))

Theoremelsymdifxor 3883 Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Theoremdfsymdif2 3884* Alternate definition of the symmetric difference. (Contributed by BJ, 30-Apr-2020.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}

Theoremsymdif2 3885* Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ ¬ (𝑥𝐴𝑥𝐵)}

Theoremsymdifass 3886 Symmetric difference associates. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 △ (𝐵𝐶)) = ((𝐴𝐵) △ 𝐶)

2.1.13.5  Combinations of difference, union, and intersection of two classes

Theoremunabs 3887 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴

Theoreminabs 3888 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∩ (𝐴𝐵)) = 𝐴

Theoremnssinpss 3889 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.)
𝐴𝐵 ↔ (𝐴𝐵) ⊊ 𝐴)

Theoremnsspssun 3890 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
𝐴𝐵𝐵 ⊊ (𝐴𝐵))

Theoremdfss4 3891 Subclass defined in terms of class difference. See comments under dfun2 3892. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴)

Theoremdfun2 3892 An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3893 and dfss4 3891 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.)
(𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Theoremdfin2 3893 An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3892. Another version is given by dfin4 3900. (Contributed by NM, 10-Jun-2004.)
(𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))

Theoremdifin 3894 Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)

Theoremssdifim 3895 Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022.)
((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))

Theoremssdifsym 3896 Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.)
((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) ↔ 𝐴 = (𝑉𝐵)))

Theoremdfss5 3897* 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.)
(𝐴𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)

Theoremdfun3 3898 Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
(𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))

Theoremdfin3 3899 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.)
(𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Theoremdfin4 3900 Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)
(𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))

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