Theorem List for Intuitionistic Logic Explorer - 3201-3300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | difeqri 3201* |
Inference from membership to difference. (Contributed by NM,
17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) |
|
Theorem | nfdif 3202 |
Bound-variable hypothesis builder for class difference. (Contributed by
NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
|
![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | eldifi 3203 |
Implication of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | eldifn 3204 |
Implication of membership in a class difference. (Contributed by NM,
3-May-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | elndif 3205 |
A set does not belong to a class excluding it. (Contributed by NM,
27-Jun-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | difdif 3206 |
Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
(Contributed by NM, 17-May-1998.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![A A](_ca.gif) |
|
Theorem | difss 3207 |
Subclass relationship for class difference. Exercise 14 of
[TakeutiZaring] p. 22.
(Contributed by NM, 29-Apr-1994.)
|
![( (](lp.gif) ![B B](_cb.gif) ![A A](_ca.gif) |
|
Theorem | difssd 3208 |
A difference of two classes is contained in the minuend. Deduction form
of difss 3207. (Contributed by David Moews, 1-May-2017.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | difss2 3209 |
If a class is contained in a difference, it is contained in the minuend.
(Contributed by David Moews, 1-May-2017.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | difss2d 3210 |
If a class is contained in a difference, it is contained in the minuend.
Deduction form of difss2 3209. (Contributed by David Moews,
1-May-2017.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | ssdifss 3211 |
Preservation of a subclass relationship by class difference. (Contributed
by NM, 15-Feb-2007.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | ddifnel 3212* |
Double complement under universal class. The hypothesis corresponds to
stability of membership in , which is weaker than decidability
(see dcstab 830). Actually, the conclusion is a
characterization of
stability of membership in a class (see ddifstab 3213) . Exercise 4.10(s)
of [Mendelson] p. 231, but with an
additional hypothesis. For a version
without a hypothesis, but which only states that is a subset of
![( (](lp.gif) ![A A](_ca.gif) , see ddifss 3319. (Contributed by Jim Kingdon,
21-Jul-2018.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![A A](_ca.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![A A](_ca.gif) |
|
Theorem | ddifstab 3213* |
A class is equal to its double complement if and only if it is stable
(that is, membership in it is a stable property). (Contributed by BJ,
12-Dec-2021.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![A. A.](forall.gif) STAB ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | ssconb 3214 |
Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif)
![B B](_cb.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | sscon 3215 |
Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
(Contributed by NM, 22-Mar-1998.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssdif 3216 |
Difference law for subsets. (Contributed by NM, 28-May-1998.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssdifd 3217 |
If is contained in
, then ![( (](lp.gif) ![C C](_cc.gif) is contained in
![( (](lp.gif) ![C C](_cc.gif) .
Deduction form of ssdif 3216. (Contributed by David
Moews, 1-May-2017.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif)
![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | sscond 3218 |
If is contained in
, then ![( (](lp.gif) ![B B](_cb.gif) is contained in
![( (](lp.gif) ![A A](_ca.gif) .
Deduction form of sscon 3215. (Contributed by David
Moews, 1-May-2017.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif)
![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssdifssd 3219 |
If is contained in
, then ![( (](lp.gif) ![C C](_cc.gif) is also contained in
. Deduction
form of ssdifss 3211. (Contributed by David Moews,
1-May-2017.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | ssdif2d 3220 |
If is contained in
and is contained in , then
![( (](lp.gif) ![D D](_cd.gif) is
contained in ![( (](lp.gif) ![C C](_cc.gif) .
Deduction form.
(Contributed by David Moews, 1-May-2017.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | raldifb 3221 |
Restricted universal quantification on a class difference in terms of an
implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
|
![( (](lp.gif) ![A. A.](forall.gif) ![( (](lp.gif) ![ph ph](_varphi.gif) ![A. A.](forall.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![ph ph](_varphi.gif) ![) )](rp.gif) |
|
2.1.13.2 The union of two classes
|
|
Theorem | elun 3222 |
Expansion of membership in class union. Theorem 12 of [Suppes] p. 25.
(Contributed by NM, 7-Aug-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | uneqri 3223* |
Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) |
|
Theorem | unidm 3224 |
Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
(Contributed by NM, 5-Aug-1993.)
|
![( (](lp.gif) ![A A](_ca.gif)
![A A](_ca.gif) |
|
Theorem | uncom 3225 |
Commutative law for union of classes. Exercise 6 of [TakeutiZaring]
p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew
Salmon, 26-Jun-2011.)
|
![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | equncom 3226 |
If a class equals the union of two other classes, then it equals the union
of those two classes commuted. (Contributed by Alan Sare,
18-Feb-2012.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | equncomi 3227 |
Inference form of equncom 3226. (Contributed by Alan Sare,
18-Feb-2012.)
|
![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | uneq1 3228 |
Equality theorem for union of two classes. (Contributed by NM,
5-Aug-1993.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | uneq2 3229 |
Equality theorem for the union of two classes. (Contributed by NM,
5-Aug-1993.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | uneq12 3230 |
Equality theorem for union of two classes. (Contributed by NM,
29-Mar-1998.)
|
![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | uneq1i 3231 |
Inference adding union to the right in a class equality. (Contributed
by NM, 30-Aug-1993.)
|
![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | uneq2i 3232 |
Inference adding union to the left in a class equality. (Contributed by
NM, 30-Aug-1993.)
|
![( (](lp.gif) ![A A](_ca.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | uneq12i 3233 |
Equality inference for union of two classes. (Contributed by NM,
12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|
![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) |
|
Theorem | uneq1d 3234 |
Deduction adding union to the right in a class equality. (Contributed
by NM, 29-Mar-1998.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | uneq2d 3235 |
Deduction adding union to the left in a class equality. (Contributed by
NM, 29-Mar-1998.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | uneq12d 3236 |
Equality deduction for union of two classes. (Contributed by NM,
29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | nfun 3237 |
Bound-variable hypothesis builder for the union of classes.
(Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro,
14-Oct-2016.)
|
![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | unass 3238 |
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.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif)
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | un12 3239 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif)
![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | un23 3240 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof
shortened by Andrew Salmon, 26-Jun-2011.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif)
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | un4 3241 |
A rearrangement of the union of 4 classes. (Contributed by NM,
12-Aug-2004.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | unundi 3242 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | unundir 3243 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif)
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssun1 3244 |
Subclass relationship for union of classes. Theorem 25 of [Suppes]
p. 27. (Contributed by NM, 5-Aug-1993.)
|
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | ssun2 3245 |
Subclass relationship for union of classes. (Contributed by NM,
30-Aug-1993.)
|
![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | ssun3 3246 |
Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
|
![( (](lp.gif)
![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssun4 3247 |
Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
|
![( (](lp.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | elun1 3248 |
Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | elun2 3249 |
Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | unss1 3250 |
Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssequn1 3251 |
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.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | unss2 3252 |
Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18.
(Contributed by NM, 14-Oct-1999.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | unss12 3253 |
Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
|
![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssequn2 3254 |
A relationship between subclass and union. (Contributed by NM,
13-Jun-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | unss 3255 |
The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27
and its converse. (Contributed by NM, 11-Jun-2004.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | unssi 3256 |
An inference showing the union of two subclasses is a subclass.
(Contributed by Raph Levien, 10-Dec-2002.)
|
![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) |
|
Theorem | unssd 3257 |
A deduction showing the union of two subclasses is a subclass.
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|
![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | unssad 3258 |
If ![( (](lp.gif) ![B B](_cb.gif) is contained
in , so is . One-way
deduction form of unss 3255. Partial converse of unssd 3257. (Contributed
by David Moews, 1-May-2017.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | unssbd 3259 |
If ![( (](lp.gif) ![B B](_cb.gif) is contained
in , so is . One-way
deduction form of unss 3255. Partial converse of unssd 3257. (Contributed
by David Moews, 1-May-2017.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | ssun 3260 |
A condition that implies inclusion in the union of two classes.
(Contributed by NM, 23-Nov-2003.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | rexun 3261 |
Restricted existential quantification over union. (Contributed by Jeff
Madsen, 5-Jan-2011.)
|
![( (](lp.gif) ![E. E.](exists.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![E. E.](exists.gif)
![E. E.](exists.gif) ![ph ph](_varphi.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ralunb 3262 |
Restricted quantification over a union. (Contributed by Scott Fenton,
12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|
![( (](lp.gif) ![A. A.](forall.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![A. A.](forall.gif)
![A. A.](forall.gif) ![ph ph](_varphi.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ralun 3263 |
Restricted quantification over union. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
![( (](lp.gif) ![( (](lp.gif) ![A. A.](forall.gif)
![A. A.](forall.gif) ![ph ph](_varphi.gif) ![A. A.](forall.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![ph ph](_varphi.gif) ![) )](rp.gif) |
|
2.1.13.3 The intersection of two
classes
|
|
Theorem | elin 3264 |
Expansion of membership in an intersection of two classes. Theorem 12
of [Suppes] p. 25. (Contributed by NM,
29-Apr-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | elini 3265 |
Membership in an intersection of two classes. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | elind 3266 |
Deduce membership in an intersection of two classes. (Contributed by
Jonathan Ben-Naim, 3-Jun-2011.)
|
![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | elinel1 3267 |
Membership in an intersection implies membership in the first set.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | elinel2 3268 |
Membership in an intersection implies membership in the second set.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | elin2 3269 |
Membership in a class defined as an intersection. (Contributed by
Stefan O'Rear, 29-Mar-2015.)
|
![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | elin1d 3270 |
Elementhood in the first set of an intersection - deduction version.
(Contributed by Thierry Arnoux, 3-May-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | elin2d 3271 |
Elementhood in the first set of an intersection - deduction version.
(Contributed by Thierry Arnoux, 3-May-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | elin3 3272 |
Membership in a class defined as a ternary intersection. (Contributed
by Stefan O'Rear, 29-Mar-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![( (](lp.gif) ![( (](lp.gif)
![D D](_cd.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | incom 3273 |
Commutative law for intersection of classes. Exercise 7 of
[TakeutiZaring] p. 17.
(Contributed by NM, 5-Aug-1993.)
|
![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | ineqri 3274* |
Inference from membership to intersection. (Contributed by NM,
5-Aug-1993.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) |
|
Theorem | ineq1 3275 |
Equality theorem for intersection of two classes. (Contributed by NM,
14-Dec-1993.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ineq2 3276 |
Equality theorem for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ineq12 3277 |
Equality theorem for intersection of two classes. (Contributed by NM,
8-May-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ineq1i 3278 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | ineq2i 3279 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | ineq12i 3280 |
Equality inference for intersection of two classes. (Contributed by
NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|
![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) |
|
Theorem | ineq1d 3281 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ineq2d 3282 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ineq12d 3283 |
Equality deduction for intersection of two classes. (Contributed by
NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ineqan12d 3284 |
Equality deduction for intersection of two classes. (Contributed by
NM, 7-Feb-2007.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![( (](lp.gif) ![ps
ps](_psi.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | dfss1 3285 |
A frequently-used variant of subclass definition df-ss 3089. (Contributed
by NM, 10-Jan-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | dfss5 3286 |
Another definition of subclasshood. Similar to df-ss 3089, dfss 3090, and
dfss1 3285. (Contributed by David Moews, 1-May-2017.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | nfin 3287 |
Bound-variable hypothesis builder for the intersection of classes.
(Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro,
14-Oct-2016.)
|
![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | csbing 3288 |
Distribute proper substitution through an intersection relation.
(Contributed by Alan Sare, 22-Jul-2012.)
|
![( (](lp.gif) ![[_
[_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![( (](lp.gif) ![D D](_cd.gif)
![( (](lp.gif) ![[_ [_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![[_ [_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | rabbi2dva 3289* |
Deduction from a wff to a restricted class abstraction. (Contributed by
NM, 14-Jan-2014.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif)
![ps ps](_psi.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![{ {](lbrace.gif) ![ps ps](_psi.gif) ![} }](rbrace.gif) ![)
)](rp.gif) |
|
Theorem | inidm 3290 |
Idempotent law for intersection of classes. Theorem 15 of [Suppes]
p. 26. (Contributed by NM, 5-Aug-1993.)
|
![( (](lp.gif) ![A A](_ca.gif)
![A A](_ca.gif) |
|
Theorem | inass 3291 |
Associative law for intersection of classes. Exercise 9 of
[TakeutiZaring] p. 17.
(Contributed by NM, 3-May-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif)
![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | in12 3292 |
A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif)
![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | in32 3293 |
A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![B
B](_cb.gif) ![) )](rp.gif) |
|
Theorem | in13 3294 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif)
![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | in31 3295 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![A
A](_ca.gif) ![) )](rp.gif) |
|
Theorem | inrot 3296 |
Rotate the intersection of 3 classes. (Contributed by NM,
27-Aug-2012.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![B
B](_cb.gif) ![) )](rp.gif) |
|
Theorem | in4 3297 |
Rearrangement of intersection of 4 classes. (Contributed by NM,
21-Apr-2001.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | inindi 3298 |
Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | inindir 3299 |
Intersection distributes over itself. (Contributed by NM,
17-Aug-2004.)
|
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | sseqin2 3300 |
A relationship between subclass and intersection. Similar to Exercise 9
of [TakeutiZaring] p. 18.
(Contributed by NM, 17-May-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![A A](_ca.gif) ![) )](rp.gif) |