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Theorem List for New Foundations Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremindi 3501 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 ∩ (BC)) = ((AB) ∪ (AC))

Theoremundi 3502 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 ∪ (BC)) = ((AB) ∩ (AC))

Theoremindir 3503 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((AB) ∩ C) = ((AC) ∪ (BC))

Theoremundir 3504 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((AB) ∪ C) = ((AC) ∩ (BC))

Theoremunineq 3505 Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)
(((AC) = (BC) (AC) = (BC)) ↔ A = B)

Theoremuneqin 3506 Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((AB) = (AB) ↔ A = B)

Theoremdifundi 3507 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(A (BC)) = ((A B) ∩ (A C))

Theoremdifundir 3508 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((AB) C) = ((A C) ∪ (B C))

Theoremdifindi 3509 Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(A (BC)) = ((A B) ∪ (A C))

Theoremdifindir 3510 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((AB) C) = ((A C) ∩ (B C))

Theoremindifdir 3511 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
((A B) ∩ C) = ((AC) (BC))

Theoremundm 3512 De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
(V (AB)) = ((V A) ∩ (V B))

Theoremindm 3513 De Morgan's law for intersection. Theorem 5.2(13') of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
(V (AB)) = ((V A) ∪ (V B))

Theoremdifun1 3514 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
(A (BC)) = ((A B) C)

Theoremundif3 3515 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.)
(A ∪ (B C)) = ((AB) (C A))

Theoremdifin2 3516 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A C → (A B) = ((C B) ∩ A))

Theoremdif32 3517 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
((A B) C) = ((A C) B)

Theoremdifabs 3518 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
((A B) B) = (A B)

Theoremsymdif1 3519 Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.)
((A B) ∪ (B A)) = ((AB) (AB))

Theoremsymdif2 3520* Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((A B) ∪ (B A)) = {x ¬ (x Ax B)}

Theoremunab 3521 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)}

Theoreminab 3522 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)}

Theoremdifab 3523 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)}

Theoremcomplab 3524 Complement of a class abstraction. (Contributed by SF, 12-Jan-2015.)
∼ {x φ} = {x ¬ φ}

Theoremnotab 3525 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
{x ¬ φ} = (V {x φ})

Theoremunrab 3526 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
({x A φ} ∪ {x A ψ}) = {x A (φ ψ)}

Theoreminrab 3527 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
({x A φ} ∩ {x A ψ}) = {x A (φ ψ)}

Theoreminrab2 3528* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
({x A φ} ∩ B) = {x (AB) φ}

Theoremdifrab 3529 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
({x A φ} {x A ψ}) = {x A (φ ¬ ψ)}

Theoremdfrab2 3530* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
{x A φ} = ({x φ} ∩ A)

Theoremdfrab3 3531* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
{x A φ} = (A ∩ {x φ})

Theoremnotrab 3532* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
(A {x A φ}) = {x A ¬ φ}

Theoremdfrab3ss 3533* 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}))

Theoremrabun2 3534 Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
{x (AB) φ} = ({x A φ} ∪ {x B φ})

Theoremreuss2 3535* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
(((A B x A (φψ)) (x A φ ∃!x B ψ)) → ∃!x A φ)

Theoremreuss 3536* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
((A B x A φ ∃!x B φ) → ∃!x A φ)

Theoremreuun1 3537* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((x A φ ∃!x (AB)(φ ψ)) → ∃!x A φ)

Theoremreuun2 3538* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
x B φ → (∃!x (AB)φ∃!x A φ))

Theoremreupick 3539* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
(((A B (x A φ ∃!x B φ)) φ) → (x Ax B))

Theoremreupick3 3540* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
((∃!x A φ x A (φ ψ) x A) → (φψ))

Theoremreupick2 3541* 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 (ψφ) x A ψ ∃!x A φ) x A) → (φψ))

Theoremsymdifcom 3542 Symmetric difference commutes. (Contributed by SF, 11-Jan-2015.)
(AB) = (BA)

Theoremcompleqb 3543 Two classes are equal iff their complements are equal. (Contributed by SF, 11-Jan-2015.)
(A = B ↔ ∼ A = ∼ B)

Theoremnecompl 3544 A class is not equal to its complement. (Contributed by SF, 11-Jan-2015.)
AA

Theoremdfin5 3545 Definition of intersection in terms of union. (Contributed by SF, 12-Jan-2015.)
(AB) = ∼ ( ∼ A ∪ ∼ B)

Theoremdfun4 3546 Definition of union in terms of intersection. (Contributed by SF, 12-Jan-2015.)
(AB) = ∼ ( ∼ A ∩ ∼ B)

Theoremiunin 3547 Intersection of two complements is equal to the complement of a union. (Contributed by SF, 12-Jan-2015.)
∼ (AB) = ( ∼ A ∩ ∼ B)

Theoremiinun 3548 Complement of intersection is equal to union of complements. (Contributed by SF, 12-Jan-2015.)
∼ (AB) = ( ∼ A ∪ ∼ B)

Theoremdifsscompl 3549 A difference is a subset of the complement of its second argument. (Contributed by SF, 10-Mar-2015.)
(A B) B

2.1.13  The empty set

Syntaxc0 3550 Extend class notation to include the empty set.
class

Definitiondf-nul 3551 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 3552. (Contributed by NM, 5-Aug-1993.)
= (V V)

Theoremdfnul2 3552 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
= {x ¬ x = x}

Theoremdfnul3 3553 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
= {x A ¬ x A}

Theoremnoel 3554 The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
¬ A

Theoremn0i 3555 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)
(B A → ¬ A = )

Theoremne0i 3556 If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)
(B AA)

Theoremvn0 3557 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
V ≠

Theoremn0f 3558 A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3559 requires only that x not be free in, rather than not occur in, A. (Contributed by NM, 17-Oct-2003.)
xA       (Ax x A)

Theoremn0 3559* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 29-Sep-2006.)
(Ax x A)

Theoremneq0 3560* A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 5-Aug-1993.)
A = x x A)

Theoremreximdva0 3561* Restricted existence deduced from non-empty class. (Contributed by NM, 1-Feb-2012.)
((φ x A) → ψ)       ((φ A) → x A ψ)

Theoremn0moeu 3562* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
(A → (∃*x x A∃!x x A))

Theoremrex0 3563 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ x φ

Theoremeq0 3564* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
(A = x ¬ x A)

Theoremeqv 3565* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
(A = V ↔ x x A)

Theorem0el 3566* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
( Ax A y ¬ y x)

Theoremabvor0 3567* The class builder of a wff not containing the abstraction variable is either the universal class or the empty set. (Contributed by Mario Carneiro, 29-Aug-2013.)
({x φ} = V {x φ} = )

Theoremabn0 3568 Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
({x φ} ≠ xφ)

Theoremab0 3569 Empty class abstraction. (Contributed by SF, 5-Jan-2018.)
({x φ} = x ¬ φ)

Theoremrabn0 3570 Non-empty restricted class abstraction. (Contributed by NM, 29-Aug-1999.)
({x A φ} ≠ x A φ)

Theoremrab0 3571 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.)
{x φ} =

Theoremrabeq0 3572 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
({x A φ} = x A ¬ φ)

Theoremrabxm 3573* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
A = ({x A φ} ∪ {x A ¬ φ})

Theoremrabnc 3574* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
({x A φ} ∩ {x A ¬ φ}) =

Theoremun0 3575 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
(A) = A

Theoremin0 3576 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
(A) =

Theoreminv1 3577 The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(A ∩ V) = A

Theoremunv 3578 The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(A ∪ V) = V

Theorem0ss 3579 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
A

Theoremss0b 3580 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
(A A = )

Theoremss0 3581 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(A A = )

Theoremsseq0 3582 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((A B B = ) → A = )

Theoremssn0 3583 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
((A B A) → B)

Theoremabf 3584 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
¬ φ       {x φ} =

Theoremeq0rdv 3585* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
(φ → ¬ x A)       (φA = )

Theoremun00 3586 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
((A = B = ) ↔ (AB) = )

Theoremvss 3587 Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(V AA = V)

Theorem0pss 3588 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
(AA)

Theoremnpss0 3589 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
¬ A

Theorempssv 3590 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
(A ⊊ V ↔ ¬ A = V)

Theoremdisj 3591* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
((AB) = x A ¬ x B)

Theoremdisjr 3592* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
((AB) = x B ¬ x A)

Theoremdisj1 3593* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
((AB) = x(x A → ¬ x B))

Theoremreldisj 3594 Two ways of saying that two classes are disjoint, using the complement of B relative to a universe C. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(A C → ((AB) = A (C B)))

Theoremdisj3 3595 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
((AB) = A = (A B))

Theoremdisjne 3596 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((AB) = C A D B) → CD)

Theoremdisjel 3597 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
(((AB) = C A) → ¬ C B)

Theoremdisj2 3598 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
((AB) = A (V B))

Theoremdisj4 3599 Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)
((AB) = ↔ ¬ (A B) ⊊ A)

Theoremssdisj 3600 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
((A B (BC) = ) → (AC) = )

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