Theorem List for Intuitionistic Logic Explorer - 3101-3200 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | difss2d 3101 |
If a class is contained in a difference, it is contained in the minuend.
Deduction form of difss2 3100. (Contributed by David Moews,
1-May-2017.)
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Theorem | ssdifss 3102 |
Preservation of a subclass relationship by class difference. (Contributed
by NM, 15-Feb-2007.)
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Theorem | ddifnel 3103* |
Double complement under universal class. The hypothesis corresponds to
stability of membership in , which is weaker than decidability
(see dcimpstab 785). Actually, the conclusion is a
characterization of
stability of membership in a class (see ddifstab 3104) . 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 3202. (Contributed by Jim Kingdon,
21-Jul-2018.)
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Theorem | ddifstab 3104* |
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 3105 |
Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
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Theorem | sscon 3106 |
Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
(Contributed by NM, 22-Mar-1998.)
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Theorem | ssdif 3107 |
Difference law for subsets. (Contributed by NM, 28-May-1998.)
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Theorem | ssdifd 3108 |
If is contained in
, then is contained in
.
Deduction form of ssdif 3107. (Contributed by David
Moews, 1-May-2017.)
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Theorem | sscond 3109 |
If is contained in
, then is contained in
.
Deduction form of sscon 3106. (Contributed by David
Moews, 1-May-2017.)
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Theorem | ssdifssd 3110 |
If is contained in
, then is also contained in
. Deduction
form of ssdifss 3102. (Contributed by David Moews,
1-May-2017.)
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Theorem | ssdif2d 3111 |
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 3112 |
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 3113 |
Expansion of membership in class union. Theorem 12 of [Suppes] p. 25.
(Contributed by NM, 7-Aug-1994.)
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Theorem | uneqri 3114* |
Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
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Theorem | unidm 3115 |
Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
(Contributed by NM, 5-Aug-1993.)
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Theorem | uncom 3116 |
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 3117 |
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 3118 |
Inference form of equncom 3117. (Contributed by Alan Sare,
18-Feb-2012.)
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Theorem | uneq1 3119 |
Equality theorem for union of two classes. (Contributed by NM,
5-Aug-1993.)
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Theorem | uneq2 3120 |
Equality theorem for the union of two classes. (Contributed by NM,
5-Aug-1993.)
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Theorem | uneq12 3121 |
Equality theorem for union of two classes. (Contributed by NM,
29-Mar-1998.)
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Theorem | uneq1i 3122 |
Inference adding union to the right in a class equality. (Contributed
by NM, 30-Aug-1993.)
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Theorem | uneq2i 3123 |
Inference adding union to the left in a class equality. (Contributed by
NM, 30-Aug-1993.)
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Theorem | uneq12i 3124 |
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 3125 |
Deduction adding union to the right in a class equality. (Contributed
by NM, 29-Mar-1998.)
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Theorem | uneq2d 3126 |
Deduction adding union to the left in a class equality. (Contributed by
NM, 29-Mar-1998.)
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Theorem | uneq12d 3127 |
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 3128 |
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 3129 |
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 3130 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
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Theorem | un23 3131 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof
shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | un4 3132 |
A rearrangement of the union of 4 classes. (Contributed by NM,
12-Aug-2004.)
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Theorem | unundi 3133 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
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Theorem | unundir 3134 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
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Theorem | ssun1 3135 |
Subclass relationship for union of classes. Theorem 25 of [Suppes]
p. 27. (Contributed by NM, 5-Aug-1993.)
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Theorem | ssun2 3136 |
Subclass relationship for union of classes. (Contributed by NM,
30-Aug-1993.)
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Theorem | ssun3 3137 |
Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
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Theorem | ssun4 3138 |
Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
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Theorem | elun1 3139 |
Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
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Theorem | elun2 3140 |
Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
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Theorem | unss1 3141 |
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 3142 |
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 3143 |
Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18.
(Contributed by NM, 14-Oct-1999.)
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Theorem | unss12 3144 |
Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
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Theorem | ssequn2 3145 |
A relationship between subclass and union. (Contributed by NM,
13-Jun-1994.)
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Theorem | unss 3146 |
The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27
and its converse. (Contributed by NM, 11-Jun-2004.)
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Theorem | unssi 3147 |
An inference showing the union of two subclasses is a subclass.
(Contributed by Raph Levien, 10-Dec-2002.)
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Theorem | unssd 3148 |
A deduction showing the union of two subclasses is a subclass.
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
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Theorem | unssad 3149 |
If is contained
in , so is . One-way
deduction form of unss 3146. Partial converse of unssd 3148. (Contributed
by David Moews, 1-May-2017.)
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Theorem | unssbd 3150 |
If is contained
in , so is . One-way
deduction form of unss 3146. Partial converse of unssd 3148. (Contributed
by David Moews, 1-May-2017.)
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Theorem | ssun 3151 |
A condition that implies inclusion in the union of two classes.
(Contributed by NM, 23-Nov-2003.)
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Theorem | rexun 3152 |
Restricted existential quantification over union. (Contributed by Jeff
Madsen, 5-Jan-2011.)
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Theorem | ralunb 3153 |
Restricted quantification over a union. (Contributed by Scott Fenton,
12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
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Theorem | ralun 3154 |
Restricted quantification over union. (Contributed by Jeff Madsen,
2-Sep-2009.)
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2.1.13.3 The intersection of two
classes
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Theorem | elin 3155 |
Expansion of membership in an intersection of two classes. Theorem 12
of [Suppes] p. 25. (Contributed by NM,
29-Apr-1994.)
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Theorem | elin2 3156 |
Membership in a class defined as an intersection. (Contributed by
Stefan O'Rear, 29-Mar-2015.)
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Theorem | elin3 3157 |
Membership in a class defined as a ternary intersection. (Contributed
by Stefan O'Rear, 29-Mar-2015.)
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Theorem | incom 3158 |
Commutative law for intersection of classes. Exercise 7 of
[TakeutiZaring] p. 17.
(Contributed by NM, 5-Aug-1993.)
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Theorem | ineqri 3159* |
Inference from membership to intersection. (Contributed by NM,
5-Aug-1993.)
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Theorem | ineq1 3160 |
Equality theorem for intersection of two classes. (Contributed by NM,
14-Dec-1993.)
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Theorem | ineq2 3161 |
Equality theorem for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
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Theorem | ineq12 3162 |
Equality theorem for intersection of two classes. (Contributed by NM,
8-May-1994.)
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Theorem | ineq1i 3163 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
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Theorem | ineq2i 3164 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
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Theorem | ineq12i 3165 |
Equality inference for intersection of two classes. (Contributed by
NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
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Theorem | ineq1d 3166 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
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Theorem | ineq2d 3167 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
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Theorem | ineq12d 3168 |
Equality deduction for intersection of two classes. (Contributed by
NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | ineqan12d 3169 |
Equality deduction for intersection of two classes. (Contributed by
NM, 7-Feb-2007.)
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Theorem | dfss1 3170 |
A frequently-used variant of subclass definition df-ss 2986. (Contributed
by NM, 10-Jan-2015.)
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Theorem | dfss5 3171 |
Another definition of subclasshood. Similar to df-ss 2986, dfss 2987, and
dfss1 3170. (Contributed by David Moews, 1-May-2017.)
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Theorem | nfin 3172 |
Bound-variable hypothesis builder for the intersection of classes.
(Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro,
14-Oct-2016.)
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Theorem | csbing 3173 |
Distribute proper substitution through an intersection relation.
(Contributed by Alan Sare, 22-Jul-2012.)
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Theorem | rabbi2dva 3174* |
Deduction from a wff to a restricted class abstraction. (Contributed by
NM, 14-Jan-2014.)
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Theorem | inidm 3175 |
Idempotent law for intersection of classes. Theorem 15 of [Suppes]
p. 26. (Contributed by NM, 5-Aug-1993.)
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Theorem | inass 3176 |
Associative law for intersection of classes. Exercise 9 of
[TakeutiZaring] p. 17.
(Contributed by NM, 3-May-1994.)
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Theorem | in12 3177 |
A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
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Theorem | in32 3178 |
A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | in13 3179 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
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Theorem | in31 3180 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
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Theorem | inrot 3181 |
Rotate the intersection of 3 classes. (Contributed by NM,
27-Aug-2012.)
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Theorem | in4 3182 |
Rearrangement of intersection of 4 classes. (Contributed by NM,
21-Apr-2001.)
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Theorem | inindi 3183 |
Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
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Theorem | inindir 3184 |
Intersection distributes over itself. (Contributed by NM,
17-Aug-2004.)
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Theorem | sseqin2 3185 |
A relationship between subclass and intersection. Similar to Exercise 9
of [TakeutiZaring] p. 18.
(Contributed by NM, 17-May-1994.)
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Theorem | inss1 3186 |
The intersection of two classes is a subset of one of them. Part of
Exercise 12 of [TakeutiZaring] p.
18. (Contributed by NM,
27-Apr-1994.)
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Theorem | inss2 3187 |
The intersection of two classes is a subset of one of them. Part of
Exercise 12 of [TakeutiZaring] p.
18. (Contributed by NM,
27-Apr-1994.)
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Theorem | ssin 3188 |
Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26.
(Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon,
26-Jun-2011.)
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Theorem | ssini 3189 |
An inference showing that a subclass of two classes is a subclass of
their intersection. (Contributed by NM, 24-Nov-2003.)
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Theorem | ssind 3190 |
A deduction showing that a subclass of two classes is a subclass of
their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
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Theorem | ssrin 3191 |
Add right intersection to subclass relation. (Contributed by NM,
16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | sslin 3192 |
Add left intersection to subclass relation. (Contributed by NM,
19-Oct-1999.)
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Theorem | ss2in 3193 |
Intersection of subclasses. (Contributed by NM, 5-May-2000.)
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Theorem | ssinss1 3194 |
Intersection preserves subclass relationship. (Contributed by NM,
14-Sep-1999.)
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Theorem | inss 3195 |
Inclusion of an intersection of two classes. (Contributed by NM,
30-Oct-2014.)
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2.1.13.4 Combinations of difference, union, and
intersection of two classes
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Theorem | unabs 3196 |
Absorption law for union. (Contributed by NM, 16-Apr-2006.)
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Theorem | inabs 3197 |
Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
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Theorem | ssddif 3198 |
Double complement and subset. Similar to ddifss 3202 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,
24-Jul-2018.)
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Theorem | unssdif 3199 |
Union of two classes and class difference. In classical logic this
would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
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Theorem | inssdif 3200 |
Intersection of two classes and class difference. In classical logic
this would be an equality. (Contributed by Jim Kingdon,
24-Jul-2018.)
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