Type  Label  Description 
Statement 

Theorem  ineq1 3301 
Equality theorem for intersection of two classes. (Contributed by NM,
14Dec1993.)



Theorem  ineq2 3302 
Equality theorem for intersection of two classes. (Contributed by NM,
26Dec1993.)



Theorem  ineq12 3303 
Equality theorem for intersection of two classes. (Contributed by NM,
8May1994.)



Theorem  ineq1i 3304 
Equality inference for intersection of two classes. (Contributed by NM,
26Dec1993.)



Theorem  ineq2i 3305 
Equality inference for intersection of two classes. (Contributed by NM,
26Dec1993.)



Theorem  ineq12i 3306 
Equality inference for intersection of two classes. (Contributed by
NM, 24Jun2004.) (Proof shortened by Eric Schmidt, 26Jan2007.)



Theorem  ineq1d 3307 
Equality deduction for intersection of two classes. (Contributed by NM,
10Apr1994.)



Theorem  ineq2d 3308 
Equality deduction for intersection of two classes. (Contributed by NM,
10Apr1994.)



Theorem  ineq12d 3309 
Equality deduction for intersection of two classes. (Contributed by
NM, 24Jun2004.) (Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  ineqan12d 3310 
Equality deduction for intersection of two classes. (Contributed by
NM, 7Feb2007.)



Theorem  dfss1 3311 
A frequentlyused variant of subclass definition dfss 3115. (Contributed
by NM, 10Jan2015.)



Theorem  dfss5 3312 
Another definition of subclasshood. Similar to dfss 3115, dfss 3116, and
dfss1 3311. (Contributed by David Moews, 1May2017.)



Theorem  nfin 3313 
Boundvariable hypothesis builder for the intersection of classes.
(Contributed by NM, 15Sep2003.) (Revised by Mario Carneiro,
14Oct2016.)



Theorem  csbing 3314 
Distribute proper substitution through an intersection relation.
(Contributed by Alan Sare, 22Jul2012.)



Theorem  rabbi2dva 3315* 
Deduction from a wff to a restricted class abstraction. (Contributed by
NM, 14Jan2014.)



Theorem  inidm 3316 
Idempotent law for intersection of classes. Theorem 15 of [Suppes]
p. 26. (Contributed by NM, 5Aug1993.)



Theorem  inass 3317 
Associative law for intersection of classes. Exercise 9 of
[TakeutiZaring] p. 17.
(Contributed by NM, 3May1994.)



Theorem  in12 3318 
A rearrangement of intersection. (Contributed by NM, 21Apr2001.)



Theorem  in32 3319 
A rearrangement of intersection. (Contributed by NM, 21Apr2001.)
(Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  in13 3320 
A rearrangement of intersection. (Contributed by NM, 27Aug2012.)



Theorem  in31 3321 
A rearrangement of intersection. (Contributed by NM, 27Aug2012.)



Theorem  inrot 3322 
Rotate the intersection of 3 classes. (Contributed by NM,
27Aug2012.)



Theorem  in4 3323 
Rearrangement of intersection of 4 classes. (Contributed by NM,
21Apr2001.)



Theorem  inindi 3324 
Intersection distributes over itself. (Contributed by NM, 6May1994.)



Theorem  inindir 3325 
Intersection distributes over itself. (Contributed by NM,
17Aug2004.)



Theorem  sseqin2 3326 
A relationship between subclass and intersection. Similar to Exercise 9
of [TakeutiZaring] p. 18.
(Contributed by NM, 17May1994.)



Theorem  inss1 3327 
The intersection of two classes is a subset of one of them. Part of
Exercise 12 of [TakeutiZaring] p.
18. (Contributed by NM,
27Apr1994.)



Theorem  inss2 3328 
The intersection of two classes is a subset of one of them. Part of
Exercise 12 of [TakeutiZaring] p.
18. (Contributed by NM,
27Apr1994.)



Theorem  ssin 3329 
Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26.
(Contributed by NM, 15Jun2004.) (Proof shortened by Andrew Salmon,
26Jun2011.)



Theorem  ssini 3330 
An inference showing that a subclass of two classes is a subclass of
their intersection. (Contributed by NM, 24Nov2003.)



Theorem  ssind 3331 
A deduction showing that a subclass of two classes is a subclass of
their intersection. (Contributed by Jonathan BenNaim, 3Jun2011.)



Theorem  ssrin 3332 
Add right intersection to subclass relation. (Contributed by NM,
16Aug1994.) (Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  sslin 3333 
Add left intersection to subclass relation. (Contributed by NM,
19Oct1999.)



Theorem  ssrind 3334 
Add right intersection to subclass relation. (Contributed by Glauco
Siliprandi, 2Jan2022.)



Theorem  ss2in 3335 
Intersection of subclasses. (Contributed by NM, 5May2000.)



Theorem  ssinss1 3336 
Intersection preserves subclass relationship. (Contributed by NM,
14Sep1999.)



Theorem  inss 3337 
Inclusion of an intersection of two classes. (Contributed by NM,
30Oct2014.)



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


Theorem  unabs 3338 
Absorption law for union. (Contributed by NM, 16Apr2006.)



Theorem  inabs 3339 
Absorption law for intersection. (Contributed by NM, 16Apr2006.)



Theorem  dfss4st 3340* 
Subclass defined in terms of class difference. (Contributed by NM,
22Mar1998.) (Proof shortened by Andrew Salmon, 26Jun2011.)

STAB


Theorem  ssddif 3341 
Double complement and subset. Similar to ddifss 3345 but inside a class
instead of the
universal class .
In classical logic the
subset operation on the right hand side could be an equality (that is,
).
(Contributed by Jim Kingdon,
24Jul2018.)



Theorem  unssdif 3342 
Union of two classes and class difference. In classical logic this
would be an equality. (Contributed by Jim Kingdon, 24Jul2018.)



Theorem  inssdif 3343 
Intersection of two classes and class difference. In classical logic
this would be an equality. (Contributed by Jim Kingdon,
24Jul2018.)



Theorem  difin 3344 
Difference with intersection. Theorem 33 of [Suppes] p. 29.
(Contributed by NM, 31Mar1998.) (Proof shortened by Andrew Salmon,
26Jun2011.)



Theorem  ddifss 3345 
Double complement under universal class. In classical logic (or given an
additional hypothesis, as in ddifnel 3238), this is equality rather than
subset. (Contributed by Jim Kingdon, 24Jul2018.)



Theorem  unssin 3346 
Union as a subset of class complement and intersection (De Morgan's
law). One direction of the definition of union in [Mendelson] p. 231.
This would be an equality, rather than subset, in classical logic.
(Contributed by Jim Kingdon, 25Jul2018.)



Theorem  inssun 3347 
Intersection in terms of class difference and union (De Morgan's law).
Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an
equality, rather than subset, in classical logic. (Contributed by Jim
Kingdon, 25Jul2018.)



Theorem  inssddif 3348 
Intersection of two classes and class difference. In classical logic,
such as Exercise 4.10(q) of [Mendelson]
p. 231, this is an equality rather
than subset. (Contributed by Jim Kingdon, 26Jul2018.)



Theorem  invdif 3349 
Intersection with universal complement. Remark in [Stoll] p. 20.
(Contributed by NM, 17Aug2004.)



Theorem  indif 3350 
Intersection with class difference. Theorem 34 of [Suppes] p. 29.
(Contributed by NM, 17Aug2004.)



Theorem  indif2 3351 
Bring an intersection in and out of a class difference. (Contributed by
Jeff Hankins, 15Jul2009.)



Theorem  indif1 3352 
Bring an intersection in and out of a class difference. (Contributed by
Mario Carneiro, 15May2015.)



Theorem  indifcom 3353 
Commutation law for intersection and difference. (Contributed by Scott
Fenton, 18Feb2013.)



Theorem  indi 3354 
Distributive law for intersection over union. Exercise 10 of
[TakeutiZaring] p. 17.
(Contributed by NM, 30Sep2002.) (Proof
shortened by Andrew Salmon, 26Jun2011.)



Theorem  undi 3355 
Distributive law for union over intersection. Exercise 11 of
[TakeutiZaring] p. 17.
(Contributed by NM, 30Sep2002.) (Proof
shortened by Andrew Salmon, 26Jun2011.)



Theorem  indir 3356 
Distributive law for intersection over union. Theorem 28 of [Suppes]
p. 27. (Contributed by NM, 30Sep2002.)



Theorem  undir 3357 
Distributive law for union over intersection. Theorem 29 of [Suppes]
p. 27. (Contributed by NM, 30Sep2002.)



Theorem  uneqin 3358 
Equality of union and intersection implies equality of their arguments.
(Contributed by NM, 16Apr2006.) (Proof shortened by Andrew Salmon,
26Jun2011.)



Theorem  difundi 3359 
Distributive law for class difference. Theorem 39 of [Suppes] p. 29.
(Contributed by NM, 17Aug2004.)



Theorem  difundir 3360 
Distributive law for class difference. (Contributed by NM,
17Aug2004.)



Theorem  difindiss 3361 
Distributive law for class difference. In classical logic, for example,
theorem 40 of [Suppes] p. 29, this is an
equality instead of subset.
(Contributed by Jim Kingdon, 26Jul2018.)



Theorem  difindir 3362 
Distributive law for class difference. (Contributed by NM,
17Aug2004.)



Theorem  indifdir 3363 
Distribute intersection over difference. (Contributed by Scott Fenton,
14Apr2011.)



Theorem  difdif2ss 3364 
Set difference with a set difference. In classical logic this would be
equality rather than subset. (Contributed by Jim Kingdon,
27Jul2018.)



Theorem  undm 3365 
De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19.
(Contributed by NM, 18Aug2004.)



Theorem  indmss 3366 
De Morgan's law for intersection. In classical logic, this would be
equality rather than subset, as in Theorem 5.2(13') of [Stoll] p. 19.
(Contributed by Jim Kingdon, 27Jul2018.)



Theorem  difun1 3367 
A relationship involving double difference and union. (Contributed by NM,
29Aug2004.)



Theorem  undif3ss 3368 
A subset relationship involving class union and class difference. In
classical logic, this would be equality rather than subset, as in the
first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by
Jim Kingdon, 28Jul2018.)



Theorem  difin2 3369 
Represent a set difference as an intersection with a larger difference.
(Contributed by Jeff Madsen, 2Sep2009.)



Theorem  dif32 3370 
Swap second and third argument of double difference. (Contributed by NM,
18Aug2004.)



Theorem  difabs 3371 
Absorptionlike law for class difference: you can remove a class only
once. (Contributed by FL, 2Aug2009.)



Theorem  symdif1 3372 
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, 17Aug2004.)



2.1.13.5 Class abstractions with difference,
union, and intersection of two classes


Theorem  symdifxor 3373* 
Expressing symmetric difference with exclusiveor or two differences.
(Contributed by Jim Kingdon, 28Jul2018.)



Theorem  unab 3374 
Union of two class abstractions. (Contributed by NM, 29Sep2002.)
(Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  inab 3375 
Intersection of two class abstractions. (Contributed by NM,
29Sep2002.) (Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  difab 3376 
Difference of two class abstractions. (Contributed by NM, 23Oct2004.)
(Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  notab 3377 
A class builder defined by a negation. (Contributed by FL,
18Sep2010.)



Theorem  unrab 3378 
Union of two restricted class abstractions. (Contributed by NM,
25Mar2004.)



Theorem  inrab 3379 
Intersection of two restricted class abstractions. (Contributed by NM,
1Sep2006.)



Theorem  inrab2 3380* 
Intersection with a restricted class abstraction. (Contributed by NM,
19Nov2007.)



Theorem  difrab 3381 
Difference of two restricted class abstractions. (Contributed by NM,
23Oct2004.)



Theorem  dfrab2 3382* 
Alternate definition of restricted class abstraction. (Contributed by
NM, 20Sep2003.)



Theorem  dfrab3 3383* 
Alternate definition of restricted class abstraction. (Contributed by
Mario Carneiro, 8Sep2013.)



Theorem  notrab 3384* 
Complementation of restricted class abstractions. (Contributed by Mario
Carneiro, 3Sep2015.)



Theorem  dfrab3ss 3385* 
Restricted class abstraction with a common superset. (Contributed by
Stefan O'Rear, 12Sep2015.) (Proof shortened by Mario Carneiro,
8Nov2015.)



Theorem  rabun2 3386 
Abstraction restricted to a union. (Contributed by Stefan O'Rear,
5Feb2015.)



2.1.13.6 Restricted uniqueness with difference,
union, and intersection


Theorem  reuss2 3387* 
Transfer uniqueness to a smaller subclass. (Contributed by NM,
20Oct2005.)



Theorem  reuss 3388* 
Transfer uniqueness to a smaller subclass. (Contributed by NM,
21Aug1999.)



Theorem  reuun1 3389* 
Transfer uniqueness to a smaller class. (Contributed by NM,
21Oct2005.)



Theorem  reuun2 3390* 
Transfer uniqueness to a smaller or larger class. (Contributed by NM,
21Oct2005.)



Theorem  reupick 3391* 
Restricted uniqueness "picks" a member of a subclass. (Contributed
by
NM, 21Aug1999.)



Theorem  reupick3 3392* 
Restricted uniqueness "picks" a member of a subclass. (Contributed
by
Mario Carneiro, 19Nov2016.)



Theorem  reupick2 3393* 
Restricted uniqueness "picks" a member of a subclass. (Contributed
by
Mario Carneiro, 15Dec2013.) (Proof shortened by Mario Carneiro,
19Nov2016.)



2.1.14 The empty set


Syntax  c0 3394 
Extend class notation to include the empty set.



Definition  dfnul 3395 
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 3396. (Contributed by NM, 5Aug1993.)



Theorem  dfnul2 3396 
Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring]
p. 20. (Contributed by NM, 26Dec1996.)



Theorem  dfnul3 3397 
Alternate definition of the empty set. (Contributed by NM,
25Mar2004.)



Theorem  noel 3398 
The empty set has no elements. Theorem 6.14 of [Quine] p. 44.
(Contributed by NM, 5Aug1993.) (Proof shortened by Mario Carneiro,
1Sep2015.)



Theorem  n0i 3399 
If a set has elements, it is not empty. A set with elements is also
inhabited, see elex2 2728. (Contributed by NM, 31Dec1993.)



Theorem  ne0i 3400 
If a set has elements, it is not empty. A set with elements is also
inhabited, see elex2 2728. (Contributed by NM, 31Dec1993.)

