Type  Label  Description 
Statement 

Theorem  elinel1 3201 
Membership in an intersection implies membership in the first set.
(Contributed by Glauco Siliprandi, 11Dec2019.)



Theorem  elinel2 3202 
Membership in an intersection implies membership in the second set.
(Contributed by Glauco Siliprandi, 11Dec2019.)



Theorem  elin2 3203 
Membership in a class defined as an intersection. (Contributed by
Stefan O'Rear, 29Mar2015.)



Theorem  elin1d 3204 
Elementhood in the first set of an intersection  deduction version.
(Contributed by Thierry Arnoux, 3May2020.)



Theorem  elin2d 3205 
Elementhood in the first set of an intersection  deduction version.
(Contributed by Thierry Arnoux, 3May2020.)



Theorem  elin3 3206 
Membership in a class defined as a ternary intersection. (Contributed
by Stefan O'Rear, 29Mar2015.)



Theorem  incom 3207 
Commutative law for intersection of classes. Exercise 7 of
[TakeutiZaring] p. 17.
(Contributed by NM, 5Aug1993.)



Theorem  ineqri 3208* 
Inference from membership to intersection. (Contributed by NM,
5Aug1993.)



Theorem  ineq1 3209 
Equality theorem for intersection of two classes. (Contributed by NM,
14Dec1993.)



Theorem  ineq2 3210 
Equality theorem for intersection of two classes. (Contributed by NM,
26Dec1993.)



Theorem  ineq12 3211 
Equality theorem for intersection of two classes. (Contributed by NM,
8May1994.)



Theorem  ineq1i 3212 
Equality inference for intersection of two classes. (Contributed by NM,
26Dec1993.)



Theorem  ineq2i 3213 
Equality inference for intersection of two classes. (Contributed by NM,
26Dec1993.)



Theorem  ineq12i 3214 
Equality inference for intersection of two classes. (Contributed by
NM, 24Jun2004.) (Proof shortened by Eric Schmidt, 26Jan2007.)



Theorem  ineq1d 3215 
Equality deduction for intersection of two classes. (Contributed by NM,
10Apr1994.)



Theorem  ineq2d 3216 
Equality deduction for intersection of two classes. (Contributed by NM,
10Apr1994.)



Theorem  ineq12d 3217 
Equality deduction for intersection of two classes. (Contributed by
NM, 24Jun2004.) (Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  ineqan12d 3218 
Equality deduction for intersection of two classes. (Contributed by
NM, 7Feb2007.)



Theorem  dfss1 3219 
A frequentlyused variant of subclass definition dfss 3026. (Contributed
by NM, 10Jan2015.)



Theorem  dfss5 3220 
Another definition of subclasshood. Similar to dfss 3026, dfss 3027, and
dfss1 3219. (Contributed by David Moews, 1May2017.)



Theorem  nfin 3221 
Boundvariable hypothesis builder for the intersection of classes.
(Contributed by NM, 15Sep2003.) (Revised by Mario Carneiro,
14Oct2016.)



Theorem  csbing 3222 
Distribute proper substitution through an intersection relation.
(Contributed by Alan Sare, 22Jul2012.)



Theorem  rabbi2dva 3223* 
Deduction from a wff to a restricted class abstraction. (Contributed by
NM, 14Jan2014.)



Theorem  inidm 3224 
Idempotent law for intersection of classes. Theorem 15 of [Suppes]
p. 26. (Contributed by NM, 5Aug1993.)



Theorem  inass 3225 
Associative law for intersection of classes. Exercise 9 of
[TakeutiZaring] p. 17.
(Contributed by NM, 3May1994.)



Theorem  in12 3226 
A rearrangement of intersection. (Contributed by NM, 21Apr2001.)



Theorem  in32 3227 
A rearrangement of intersection. (Contributed by NM, 21Apr2001.)
(Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  in13 3228 
A rearrangement of intersection. (Contributed by NM, 27Aug2012.)



Theorem  in31 3229 
A rearrangement of intersection. (Contributed by NM, 27Aug2012.)



Theorem  inrot 3230 
Rotate the intersection of 3 classes. (Contributed by NM,
27Aug2012.)



Theorem  in4 3231 
Rearrangement of intersection of 4 classes. (Contributed by NM,
21Apr2001.)



Theorem  inindi 3232 
Intersection distributes over itself. (Contributed by NM, 6May1994.)



Theorem  inindir 3233 
Intersection distributes over itself. (Contributed by NM,
17Aug2004.)



Theorem  sseqin2 3234 
A relationship between subclass and intersection. Similar to Exercise 9
of [TakeutiZaring] p. 18.
(Contributed by NM, 17May1994.)



Theorem  inss1 3235 
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 3236 
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 3237 
Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26.
(Contributed by NM, 15Jun2004.) (Proof shortened by Andrew Salmon,
26Jun2011.)



Theorem  ssini 3238 
An inference showing that a subclass of two classes is a subclass of
their intersection. (Contributed by NM, 24Nov2003.)



Theorem  ssind 3239 
A deduction showing that a subclass of two classes is a subclass of
their intersection. (Contributed by Jonathan BenNaim, 3Jun2011.)



Theorem  ssrin 3240 
Add right intersection to subclass relation. (Contributed by NM,
16Aug1994.) (Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  sslin 3241 
Add left intersection to subclass relation. (Contributed by NM,
19Oct1999.)



Theorem  ssrind 3242 
Add right intersection to subclass relation. (Contributed by Glauco
Siliprandi, 2Jan2022.)



Theorem  ss2in 3243 
Intersection of subclasses. (Contributed by NM, 5May2000.)



Theorem  ssinss1 3244 
Intersection preserves subclass relationship. (Contributed by NM,
14Sep1999.)



Theorem  inss 3245 
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 3246 
Absorption law for union. (Contributed by NM, 16Apr2006.)



Theorem  inabs 3247 
Absorption law for intersection. (Contributed by NM, 16Apr2006.)



Theorem  dfss4st 3248* 
Subclass defined in terms of class difference. (Contributed by NM,
22Mar1998.) (Proof shortened by Andrew Salmon, 26Jun2011.)

STAB


Theorem  ssddif 3249 
Double complement and subset. Similar to ddifss 3253 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 3250 
Union of two classes and class difference. In classical logic this
would be an equality. (Contributed by Jim Kingdon, 24Jul2018.)



Theorem  inssdif 3251 
Intersection of two classes and class difference. In classical logic
this would be an equality. (Contributed by Jim Kingdon,
24Jul2018.)



Theorem  difin 3252 
Difference with intersection. Theorem 33 of [Suppes] p. 29.
(Contributed by NM, 31Mar1998.) (Proof shortened by Andrew Salmon,
26Jun2011.)



Theorem  ddifss 3253 
Double complement under universal class. In classical logic (or given an
additional hypothesis, as in ddifnel 3146), this is equality rather than
subset. (Contributed by Jim Kingdon, 24Jul2018.)



Theorem  unssin 3254 
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 3255 
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 3256 
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 3257 
Intersection with universal complement. Remark in [Stoll] p. 20.
(Contributed by NM, 17Aug2004.)



Theorem  indif 3258 
Intersection with class difference. Theorem 34 of [Suppes] p. 29.
(Contributed by NM, 17Aug2004.)



Theorem  indif2 3259 
Bring an intersection in and out of a class difference. (Contributed by
Jeff Hankins, 15Jul2009.)



Theorem  indif1 3260 
Bring an intersection in and out of a class difference. (Contributed by
Mario Carneiro, 15May2015.)



Theorem  indifcom 3261 
Commutation law for intersection and difference. (Contributed by Scott
Fenton, 18Feb2013.)



Theorem  indi 3262 
Distributive law for intersection over union. Exercise 10 of
[TakeutiZaring] p. 17.
(Contributed by NM, 30Sep2002.) (Proof
shortened by Andrew Salmon, 26Jun2011.)



Theorem  undi 3263 
Distributive law for union over intersection. Exercise 11 of
[TakeutiZaring] p. 17.
(Contributed by NM, 30Sep2002.) (Proof
shortened by Andrew Salmon, 26Jun2011.)



Theorem  indir 3264 
Distributive law for intersection over union. Theorem 28 of [Suppes]
p. 27. (Contributed by NM, 30Sep2002.)



Theorem  undir 3265 
Distributive law for union over intersection. Theorem 29 of [Suppes]
p. 27. (Contributed by NM, 30Sep2002.)



Theorem  uneqin 3266 
Equality of union and intersection implies equality of their arguments.
(Contributed by NM, 16Apr2006.) (Proof shortened by Andrew Salmon,
26Jun2011.)



Theorem  difundi 3267 
Distributive law for class difference. Theorem 39 of [Suppes] p. 29.
(Contributed by NM, 17Aug2004.)



Theorem  difundir 3268 
Distributive law for class difference. (Contributed by NM,
17Aug2004.)



Theorem  difindiss 3269 
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 3270 
Distributive law for class difference. (Contributed by NM,
17Aug2004.)



Theorem  indifdir 3271 
Distribute intersection over difference. (Contributed by Scott Fenton,
14Apr2011.)



Theorem  difdif2ss 3272 
Set difference with a set difference. In classical logic this would be
equality rather than subset. (Contributed by Jim Kingdon,
27Jul2018.)



Theorem  undm 3273 
De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19.
(Contributed by NM, 18Aug2004.)



Theorem  indmss 3274 
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 3275 
A relationship involving double difference and union. (Contributed by NM,
29Aug2004.)



Theorem  undif3ss 3276 
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 3277 
Represent a set difference as an intersection with a larger difference.
(Contributed by Jeff Madsen, 2Sep2009.)



Theorem  dif32 3278 
Swap second and third argument of double difference. (Contributed by NM,
18Aug2004.)



Theorem  difabs 3279 
Absorptionlike law for class difference: you can remove a class only
once. (Contributed by FL, 2Aug2009.)



Theorem  symdif1 3280 
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 3281* 
Expressing symmetric difference with exclusiveor or two differences.
(Contributed by Jim Kingdon, 28Jul2018.)



Theorem  unab 3282 
Union of two class abstractions. (Contributed by NM, 29Sep2002.)
(Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  inab 3283 
Intersection of two class abstractions. (Contributed by NM,
29Sep2002.) (Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  difab 3284 
Difference of two class abstractions. (Contributed by NM, 23Oct2004.)
(Proof shortened by Andrew Salmon, 26Jun2011.)



Theorem  notab 3285 
A class builder defined by a negation. (Contributed by FL,
18Sep2010.)



Theorem  unrab 3286 
Union of two restricted class abstractions. (Contributed by NM,
25Mar2004.)



Theorem  inrab 3287 
Intersection of two restricted class abstractions. (Contributed by NM,
1Sep2006.)



Theorem  inrab2 3288* 
Intersection with a restricted class abstraction. (Contributed by NM,
19Nov2007.)



Theorem  difrab 3289 
Difference of two restricted class abstractions. (Contributed by NM,
23Oct2004.)



Theorem  dfrab2 3290* 
Alternate definition of restricted class abstraction. (Contributed by
NM, 20Sep2003.)



Theorem  dfrab3 3291* 
Alternate definition of restricted class abstraction. (Contributed by
Mario Carneiro, 8Sep2013.)



Theorem  notrab 3292* 
Complementation of restricted class abstractions. (Contributed by Mario
Carneiro, 3Sep2015.)



Theorem  dfrab3ss 3293* 
Restricted class abstraction with a common superset. (Contributed by
Stefan O'Rear, 12Sep2015.) (Proof shortened by Mario Carneiro,
8Nov2015.)



Theorem  rabun2 3294 
Abstraction restricted to a union. (Contributed by Stefan O'Rear,
5Feb2015.)



2.1.13.6 Restricted uniqueness with difference,
union, and intersection


Theorem  reuss2 3295* 
Transfer uniqueness to a smaller subclass. (Contributed by NM,
20Oct2005.)



Theorem  reuss 3296* 
Transfer uniqueness to a smaller subclass. (Contributed by NM,
21Aug1999.)



Theorem  reuun1 3297* 
Transfer uniqueness to a smaller class. (Contributed by NM,
21Oct2005.)



Theorem  reuun2 3298* 
Transfer uniqueness to a smaller or larger class. (Contributed by NM,
21Oct2005.)



Theorem  reupick 3299* 
Restricted uniqueness "picks" a member of a subclass. (Contributed
by
NM, 21Aug1999.)



Theorem  reupick3 3300* 
Restricted uniqueness "picks" a member of a subclass. (Contributed
by
Mario Carneiro, 19Nov2016.)

