Theorem List for Intuitionistic Logic Explorer - 3201-3300 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | eqimss 3201 |
Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.)
(Proof shortened by Andrew Salmon, 21-Jun-2011.)
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Theorem | eqimss2 3202 |
Equality implies the subclass relation. (Contributed by NM,
23-Nov-2003.)
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Theorem | eqimssi 3203 |
Infer subclass relationship from equality. (Contributed by NM,
6-Jan-2007.)
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Theorem | eqimss2i 3204 |
Infer subclass relationship from equality. (Contributed by NM,
7-Jan-2007.)
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Theorem | nssne1 3205 |
Two classes are different if they don't include the same class.
(Contributed by NM, 23-Apr-2015.)
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Theorem | nssne2 3206 |
Two classes are different if they are not subclasses of the same class.
(Contributed by NM, 23-Apr-2015.)
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Theorem | nssr 3207* |
Negation of subclass relationship. One direction of Exercise 13 of
[TakeutiZaring] p. 18.
(Contributed by Jim Kingdon, 15-Jul-2018.)
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Theorem | nelss 3208 |
Demonstrate by witnesses that two classes lack a subclass relation.
(Contributed by Stefan O'Rear, 5-Feb-2015.)
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Theorem | ssrexf 3209 |
Restricted existential quantification follows from a subclass
relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
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Theorem | ssrmof 3210 |
"At most one" existential quantification restricted to a subclass.
(Contributed by Thierry Arnoux, 8-Oct-2017.)
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Theorem | ssralv 3211* |
Quantification restricted to a subclass. (Contributed by NM,
11-Mar-2006.)
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Theorem | ssrexv 3212* |
Existential quantification restricted to a subclass. (Contributed by
NM, 11-Jan-2007.)
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Theorem | ralss 3213* |
Restricted universal quantification on a subset in terms of superset.
(Contributed by Stefan O'Rear, 3-Apr-2015.)
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Theorem | rexss 3214* |
Restricted existential quantification on a subset in terms of superset.
(Contributed by Stefan O'Rear, 3-Apr-2015.)
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Theorem | ss2ab 3215 |
Class abstractions in a subclass relationship. (Contributed by NM,
3-Jul-1994.)
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Theorem | abss 3216* |
Class abstraction in a subclass relationship. (Contributed by NM,
16-Aug-2006.)
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Theorem | ssab 3217* |
Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
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Theorem | ssabral 3218* |
The relation for a subclass of a class abstraction is equivalent to
restricted quantification. (Contributed by NM, 6-Sep-2006.)
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Theorem | ss2abi 3219 |
Inference of abstraction subclass from implication. (Contributed by NM,
31-Mar-1995.)
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Theorem | ss2abdv 3220* |
Deduction of abstraction subclass from implication. (Contributed by NM,
29-Jul-2011.)
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Theorem | abssdv 3221* |
Deduction of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
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Theorem | abssi 3222* |
Inference of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
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Theorem | ss2rab 3223 |
Restricted abstraction classes in a subclass relationship. (Contributed
by NM, 30-May-1999.)
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Theorem | rabss 3224* |
Restricted class abstraction in a subclass relationship. (Contributed
by NM, 16-Aug-2006.)
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Theorem | ssrab 3225* |
Subclass of a restricted class abstraction. (Contributed by NM,
16-Aug-2006.)
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Theorem | ssrabdv 3226* |
Subclass of a restricted class abstraction (deduction form).
(Contributed by NM, 31-Aug-2006.)
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Theorem | rabssdv 3227* |
Subclass of a restricted class abstraction (deduction form).
(Contributed by NM, 2-Feb-2015.)
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Theorem | ss2rabdv 3228* |
Deduction of restricted abstraction subclass from implication.
(Contributed by NM, 30-May-2006.)
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Theorem | ss2rabi 3229 |
Inference of restricted abstraction subclass from implication.
(Contributed by NM, 14-Oct-1999.)
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Theorem | rabss2 3230* |
Subclass law for restricted abstraction. (Contributed by NM,
18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | ssab2 3231* |
Subclass relation for the restriction of a class abstraction.
(Contributed by NM, 31-Mar-1995.)
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Theorem | ssrab2 3232* |
Subclass relation for a restricted class. (Contributed by NM,
19-Mar-1997.)
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Theorem | ssrab3 3233* |
Subclass relation for a restricted class abstraction. (Contributed by
Jonathan Ben-Naim, 3-Jun-2011.)
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Theorem | ssrabeq 3234* |
If the restricting class of a restricted class abstraction is a subset
of this restricted class abstraction, it is equal to this restricted
class abstraction. (Contributed by Alexander van der Vekens,
31-Dec-2017.)
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Theorem | rabssab 3235 |
A restricted class is a subclass of the corresponding unrestricted class.
(Contributed by Mario Carneiro, 23-Dec-2016.)
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Theorem | uniiunlem 3236* |
A subset relationship useful for converting union to indexed union using
dfiun2 or dfiun2g and intersection to indexed intersection using
dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario
Carneiro, 26-Sep-2015.)
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2.1.13 The difference, union, and intersection
of two classes
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2.1.13.1 The difference of two
classes
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Theorem | dfdif3 3237* |
Alternate definition of class difference. Definition of relative set
complement in Section 2.3 of [Pierik], p.
10. (Contributed by BJ and
Jim Kingdon, 16-Jun-2022.)
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Theorem | difeq1 3238 |
Equality theorem for class difference. (Contributed by NM,
10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | difeq2 3239 |
Equality theorem for class difference. (Contributed by NM,
10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | difeq12 3240 |
Equality theorem for class difference. (Contributed by FL,
31-Aug-2009.)
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Theorem | difeq1i 3241 |
Inference adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq2i 3242 |
Inference adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq12i 3243 |
Equality inference for class difference. (Contributed by NM,
29-Aug-2004.)
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Theorem | difeq1d 3244 |
Deduction adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq2d 3245 |
Deduction adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq12d 3246 |
Equality deduction for class difference. (Contributed by FL,
29-May-2014.)
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Theorem | difeqri 3247* |
Inference from membership to difference. (Contributed by NM,
17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | nfdif 3248 |
Bound-variable hypothesis builder for class difference. (Contributed by
NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
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Theorem | eldifi 3249 |
Implication of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
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Theorem | eldifn 3250 |
Implication of membership in a class difference. (Contributed by NM,
3-May-1994.)
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Theorem | elndif 3251 |
A set does not belong to a class excluding it. (Contributed by NM,
27-Jun-1994.)
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Theorem | difdif 3252 |
Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
(Contributed by NM, 17-May-1998.)
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Theorem | difss 3253 |
Subclass relationship for class difference. Exercise 14 of
[TakeutiZaring] p. 22.
(Contributed by NM, 29-Apr-1994.)
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Theorem | difssd 3254 |
A difference of two classes is contained in the minuend. Deduction form
of difss 3253. (Contributed by David Moews, 1-May-2017.)
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Theorem | difss2 3255 |
If a class is contained in a difference, it is contained in the minuend.
(Contributed by David Moews, 1-May-2017.)
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Theorem | difss2d 3256 |
If a class is contained in a difference, it is contained in the minuend.
Deduction form of difss2 3255. (Contributed by David Moews,
1-May-2017.)
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Theorem | ssdifss 3257 |
Preservation of a subclass relationship by class difference. (Contributed
by NM, 15-Feb-2007.)
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Theorem | ddifnel 3258* |
Double complement under universal class. The hypothesis corresponds to
stability of membership in , which is weaker than decidability
(see dcstab 839). Actually, the conclusion is a
characterization of
stability of membership in a class (see ddifstab 3259) . 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
, see ddifss 3365. (Contributed by Jim Kingdon,
21-Jul-2018.)
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Theorem | ddifstab 3259* |
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.)
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STAB |
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Theorem | ssconb 3260 |
Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
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Theorem | sscon 3261 |
Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
(Contributed by NM, 22-Mar-1998.)
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Theorem | ssdif 3262 |
Difference law for subsets. (Contributed by NM, 28-May-1998.)
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Theorem | ssdifd 3263 |
If is contained in
, then is contained in
.
Deduction form of ssdif 3262. (Contributed by David
Moews, 1-May-2017.)
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Theorem | sscond 3264 |
If is contained in
, then is contained in
.
Deduction form of sscon 3261. (Contributed by David
Moews, 1-May-2017.)
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Theorem | ssdifssd 3265 |
If is contained in
, then is also contained in
. Deduction
form of ssdifss 3257. (Contributed by David Moews,
1-May-2017.)
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Theorem | ssdif2d 3266 |
If is contained in
and is contained in , then
is
contained in .
Deduction form.
(Contributed by David Moews, 1-May-2017.)
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Theorem | raldifb 3267 |
Restricted universal quantification on a class difference in terms of an
implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
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2.1.13.2 The union of two classes
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Theorem | elun 3268 |
Expansion of membership in class union. Theorem 12 of [Suppes] p. 25.
(Contributed by NM, 7-Aug-1994.)
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Theorem | uneqri 3269* |
Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
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Theorem | unidm 3270 |
Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
(Contributed by NM, 5-Aug-1993.)
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Theorem | uncom 3271 |
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.)
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Theorem | equncom 3272 |
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.)
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Theorem | equncomi 3273 |
Inference form of equncom 3272. (Contributed by Alan Sare,
18-Feb-2012.)
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Theorem | uneq1 3274 |
Equality theorem for union of two classes. (Contributed by NM,
5-Aug-1993.)
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Theorem | uneq2 3275 |
Equality theorem for the union of two classes. (Contributed by NM,
5-Aug-1993.)
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Theorem | uneq12 3276 |
Equality theorem for union of two classes. (Contributed by NM,
29-Mar-1998.)
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Theorem | uneq1i 3277 |
Inference adding union to the right in a class equality. (Contributed
by NM, 30-Aug-1993.)
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Theorem | uneq2i 3278 |
Inference adding union to the left in a class equality. (Contributed by
NM, 30-Aug-1993.)
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Theorem | uneq12i 3279 |
Equality inference for union of two classes. (Contributed by NM,
12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
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Theorem | uneq1d 3280 |
Deduction adding union to the right in a class equality. (Contributed
by NM, 29-Mar-1998.)
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Theorem | uneq2d 3281 |
Deduction adding union to the left in a class equality. (Contributed by
NM, 29-Mar-1998.)
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Theorem | uneq12d 3282 |
Equality deduction for union of two classes. (Contributed by NM,
29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | nfun 3283 |
Bound-variable hypothesis builder for the union of classes.
(Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro,
14-Oct-2016.)
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Theorem | unass 3284 |
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.)
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Theorem | un12 3285 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
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Theorem | un23 3286 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof
shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | un4 3287 |
A rearrangement of the union of 4 classes. (Contributed by NM,
12-Aug-2004.)
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Theorem | unundi 3288 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
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Theorem | unundir 3289 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
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Theorem | ssun1 3290 |
Subclass relationship for union of classes. Theorem 25 of [Suppes]
p. 27. (Contributed by NM, 5-Aug-1993.)
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Theorem | ssun2 3291 |
Subclass relationship for union of classes. (Contributed by NM,
30-Aug-1993.)
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Theorem | ssun3 3292 |
Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
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Theorem | ssun4 3293 |
Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
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Theorem | elun1 3294 |
Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
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Theorem | elun2 3295 |
Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
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Theorem | unss1 3296 |
Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | ssequn1 3297 |
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.)
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Theorem | unss2 3298 |
Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18.
(Contributed by NM, 14-Oct-1999.)
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Theorem | unss12 3299 |
Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
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Theorem | ssequn2 3300 |
A relationship between subclass and union. (Contributed by NM,
13-Jun-1994.)
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