Theorem List for Intuitionistic Logic Explorer - 3101-3200 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | sseqtri 3101 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 28-Jul-1995.)
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Theorem | sseqtrri 3102 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 4-Apr-1995.)
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Theorem | eqsstrd 3103 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | eqsstrrd 3104 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtrd 3105 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtrrd 3106 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | 3sstr3i 3107 |
Substitution of equality in both sides of a subclass relationship.
(Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt,
26-Jan-2007.)
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Theorem | 3sstr4i 3108 |
Substitution of equality in both sides of a subclass relationship.
(Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt,
26-Jan-2007.)
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Theorem | 3sstr3g 3109 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4g 3110 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt,
26-Jan-2007.)
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Theorem | 3sstr3d 3111 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4d 3112 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt,
26-Jan-2007.)
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Theorem | eqsstrid 3113 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | eqsstrrid 3114 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | sseqtrdi 3115 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | sseqtrrdi 3116 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | sseqtrid 3117 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | sseqtrrid 3118 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | eqsstrdi 3119 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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Theorem | eqsstrrdi 3120 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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Theorem | eqimss 3121 |
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 3122 |
Equality implies the subclass relation. (Contributed by NM,
23-Nov-2003.)
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Theorem | eqimssi 3123 |
Infer subclass relationship from equality. (Contributed by NM,
6-Jan-2007.)
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Theorem | eqimss2i 3124 |
Infer subclass relationship from equality. (Contributed by NM,
7-Jan-2007.)
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Theorem | nssne1 3125 |
Two classes are different if they don't include the same class.
(Contributed by NM, 23-Apr-2015.)
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Theorem | nssne2 3126 |
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 3127* |
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 3128 |
Demonstrate by witnesses that two classes lack a subclass relation.
(Contributed by Stefan O'Rear, 5-Feb-2015.)
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Theorem | ssrexf 3129 |
Restricted existential quantification follows from a subclass
relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
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Theorem | ssrmof 3130 |
"At most one" existential quantification restricted to a subclass.
(Contributed by Thierry Arnoux, 8-Oct-2017.)
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Theorem | ssralv 3131* |
Quantification restricted to a subclass. (Contributed by NM,
11-Mar-2006.)
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Theorem | ssrexv 3132* |
Existential quantification restricted to a subclass. (Contributed by
NM, 11-Jan-2007.)
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Theorem | ralss 3133* |
Restricted universal quantification on a subset in terms of superset.
(Contributed by Stefan O'Rear, 3-Apr-2015.)
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Theorem | rexss 3134* |
Restricted existential quantification on a subset in terms of superset.
(Contributed by Stefan O'Rear, 3-Apr-2015.)
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Theorem | ss2ab 3135 |
Class abstractions in a subclass relationship. (Contributed by NM,
3-Jul-1994.)
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Theorem | abss 3136* |
Class abstraction in a subclass relationship. (Contributed by NM,
16-Aug-2006.)
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Theorem | ssab 3137* |
Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
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Theorem | ssabral 3138* |
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 3139 |
Inference of abstraction subclass from implication. (Contributed by NM,
31-Mar-1995.)
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Theorem | ss2abdv 3140* |
Deduction of abstraction subclass from implication. (Contributed by NM,
29-Jul-2011.)
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Theorem | abssdv 3141* |
Deduction of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
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Theorem | abssi 3142* |
Inference of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
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Theorem | ss2rab 3143 |
Restricted abstraction classes in a subclass relationship. (Contributed
by NM, 30-May-1999.)
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Theorem | rabss 3144* |
Restricted class abstraction in a subclass relationship. (Contributed
by NM, 16-Aug-2006.)
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Theorem | ssrab 3145* |
Subclass of a restricted class abstraction. (Contributed by NM,
16-Aug-2006.)
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Theorem | ssrabdv 3146* |
Subclass of a restricted class abstraction (deduction form).
(Contributed by NM, 31-Aug-2006.)
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Theorem | rabssdv 3147* |
Subclass of a restricted class abstraction (deduction form).
(Contributed by NM, 2-Feb-2015.)
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Theorem | ss2rabdv 3148* |
Deduction of restricted abstraction subclass from implication.
(Contributed by NM, 30-May-2006.)
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Theorem | ss2rabi 3149 |
Inference of restricted abstraction subclass from implication.
(Contributed by NM, 14-Oct-1999.)
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Theorem | rabss2 3150* |
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 3151* |
Subclass relation for the restriction of a class abstraction.
(Contributed by NM, 31-Mar-1995.)
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Theorem | ssrab2 3152* |
Subclass relation for a restricted class. (Contributed by NM,
19-Mar-1997.)
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Theorem | ssrabeq 3153* |
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 3154 |
A restricted class is a subclass of the corresponding unrestricted class.
(Contributed by Mario Carneiro, 23-Dec-2016.)
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Theorem | uniiunlem 3155* |
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 3156* |
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 3157 |
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 3158 |
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 3159 |
Equality theorem for class difference. (Contributed by FL,
31-Aug-2009.)
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Theorem | difeq1i 3160 |
Inference adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq2i 3161 |
Inference adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq12i 3162 |
Equality inference for class difference. (Contributed by NM,
29-Aug-2004.)
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Theorem | difeq1d 3163 |
Deduction adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq2d 3164 |
Deduction adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq12d 3165 |
Equality deduction for class difference. (Contributed by FL,
29-May-2014.)
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Theorem | difeqri 3166* |
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 3167 |
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 3168 |
Implication of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
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Theorem | eldifn 3169 |
Implication of membership in a class difference. (Contributed by NM,
3-May-1994.)
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Theorem | elndif 3170 |
A set does not belong to a class excluding it. (Contributed by NM,
27-Jun-1994.)
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Theorem | difdif 3171 |
Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
(Contributed by NM, 17-May-1998.)
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Theorem | difss 3172 |
Subclass relationship for class difference. Exercise 14 of
[TakeutiZaring] p. 22.
(Contributed by NM, 29-Apr-1994.)
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Theorem | difssd 3173 |
A difference of two classes is contained in the minuend. Deduction form
of difss 3172. (Contributed by David Moews, 1-May-2017.)
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Theorem | difss2 3174 |
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 3175 |
If a class is contained in a difference, it is contained in the minuend.
Deduction form of difss2 3174. (Contributed by David Moews,
1-May-2017.)
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Theorem | ssdifss 3176 |
Preservation of a subclass relationship by class difference. (Contributed
by NM, 15-Feb-2007.)
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Theorem | ddifnel 3177* |
Double complement under universal class. The hypothesis corresponds to
stability of membership in , which is weaker than decidability
(see dcstab 814). Actually, the conclusion is a
characterization of
stability of membership in a class (see ddifstab 3178) . 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 3284. (Contributed by Jim Kingdon,
21-Jul-2018.)
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Theorem | ddifstab 3178* |
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 3179 |
Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
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Theorem | sscon 3180 |
Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
(Contributed by NM, 22-Mar-1998.)
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Theorem | ssdif 3181 |
Difference law for subsets. (Contributed by NM, 28-May-1998.)
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Theorem | ssdifd 3182 |
If is contained in
, then is contained in
.
Deduction form of ssdif 3181. (Contributed by David
Moews, 1-May-2017.)
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Theorem | sscond 3183 |
If is contained in
, then is contained in
.
Deduction form of sscon 3180. (Contributed by David
Moews, 1-May-2017.)
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Theorem | ssdifssd 3184 |
If is contained in
, then is also contained in
. Deduction
form of ssdifss 3176. (Contributed by David Moews,
1-May-2017.)
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Theorem | ssdif2d 3185 |
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 3186 |
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 3187 |
Expansion of membership in class union. Theorem 12 of [Suppes] p. 25.
(Contributed by NM, 7-Aug-1994.)
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Theorem | uneqri 3188* |
Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
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Theorem | unidm 3189 |
Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
(Contributed by NM, 5-Aug-1993.)
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Theorem | uncom 3190 |
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 3191 |
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 3192 |
Inference form of equncom 3191. (Contributed by Alan Sare,
18-Feb-2012.)
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Theorem | uneq1 3193 |
Equality theorem for union of two classes. (Contributed by NM,
5-Aug-1993.)
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Theorem | uneq2 3194 |
Equality theorem for the union of two classes. (Contributed by NM,
5-Aug-1993.)
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Theorem | uneq12 3195 |
Equality theorem for union of two classes. (Contributed by NM,
29-Mar-1998.)
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Theorem | uneq1i 3196 |
Inference adding union to the right in a class equality. (Contributed
by NM, 30-Aug-1993.)
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Theorem | uneq2i 3197 |
Inference adding union to the left in a class equality. (Contributed by
NM, 30-Aug-1993.)
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Theorem | uneq12i 3198 |
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 3199 |
Deduction adding union to the right in a class equality. (Contributed
by NM, 29-Mar-1998.)
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Theorem | uneq2d 3200 |
Deduction adding union to the left in a class equality. (Contributed by
NM, 29-Mar-1998.)
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