Theorem List for Intuitionistic Logic Explorer - 3101-3200 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | csbabg 3101* |
Move substitution into a class abstraction. (Contributed by NM,
13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
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Theorem | cbvralcsf 3102 |
A more general version of cbvralf 2682 that doesn't require and
to be distinct from or . Changes
bound variables using
implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
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Theorem | cbvrexcsf 3103 |
A more general version of cbvrexf 2683 that has no distinct variable
restrictions. Changes bound variables using implicit substitution.
(Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario
Carneiro, 7-Dec-2014.)
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Theorem | cbvreucsf 3104 |
A more general version of cbvreuv 2691 that has no distinct variable
rextrictions. Changes bound variables using implicit substitution.
(Contributed by Andrew Salmon, 13-Jul-2011.)
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Theorem | cbvrabcsf 3105 |
A more general version of cbvrab 2719 with no distinct variable
restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
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Theorem | cbvralv2 3106* |
Rule used to change the bound variable in a restricted universal
quantifier with implicit substitution which also changes the quantifier
domain. (Contributed by David Moews, 1-May-2017.)
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Theorem | cbvrexv2 3107* |
Rule used to change the bound variable in a restricted existential
quantifier with implicit substitution which also changes the quantifier
domain. (Contributed by David Moews, 1-May-2017.)
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2.1.11 Define basic set operations and
relations
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Syntax | cdif 3108 |
Extend class notation to include class difference (read: " minus
").
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Syntax | cun 3109 |
Extend class notation to include union of two classes (read: "
union ").
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Syntax | cin 3110 |
Extend class notation to include the intersection of two classes (read:
" intersect
").
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Syntax | wss 3111 |
Extend wff notation to include the subclass relation. This is
read " is a
subclass of " or
" includes ." When
exists as a set,
it is also read "
is a subset of ."
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Theorem | difjust 3112* |
Soundness justification theorem for df-dif 3113. (Contributed by Rodolfo
Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Definition | df-dif 3113* |
Define class difference, also called relative complement. Definition
5.12 of [TakeutiZaring] p. 20.
Contrast this operation with union
(df-un 3115) and intersection (df-in 3117).
Several notations are used in the literature; we chose the
convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the
more common minus sign to reserve the latter for later use in, e.g.,
arithmetic. We will use the terminology " excludes " to
mean . We will use " is removed from " to mean
i.e. the removal of an element or equivalently the
exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)
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Theorem | unjust 3114* |
Soundness justification theorem for df-un 3115. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Definition | df-un 3115* |
Define the union of two classes. Definition 5.6 of [TakeutiZaring]
p. 16. Contrast this operation with difference
(df-dif 3113) and intersection (df-in 3117). (Contributed
by NM, 23-Aug-1993.)
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Theorem | injust 3116* |
Soundness justification theorem for df-in 3117. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Definition | df-in 3117* |
Define the intersection of two classes. Definition 5.6 of
[TakeutiZaring] p. 16. Contrast
this operation with union
(df-un 3115) and difference (df-dif 3113).
(Contributed by NM, 29-Apr-1994.)
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Theorem | dfin5 3118* |
Alternate definition for the intersection of two classes. (Contributed
by NM, 6-Jul-2005.)
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Theorem | dfdif2 3119* |
Alternate definition of class difference. (Contributed by NM,
25-Mar-2004.)
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Theorem | eldif 3120 |
Expansion of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
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Theorem | eldifd 3121 |
If a class is in one class and not another, it is also in their
difference. One-way deduction form of eldif 3120. (Contributed by David
Moews, 1-May-2017.)
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Theorem | eldifad 3122 |
If a class is in the difference of two classes, it is also in the
minuend. One-way deduction form of eldif 3120. (Contributed by David
Moews, 1-May-2017.)
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Theorem | eldifbd 3123 |
If a class is in the difference of two classes, it is not in the
subtrahend. One-way deduction form of eldif 3120. (Contributed by David
Moews, 1-May-2017.)
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2.1.12 Subclasses and subsets
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Definition | df-ss 3124 |
Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18.
Note that (proved in ssid 3157). For a more traditional
definition, but requiring a dummy variable, see dfss2 3126. Other possible
definitions are given by dfss3 3127, ssequn1 3287, ssequn2 3290, and sseqin2 3336.
(Contributed by NM, 27-Apr-1994.)
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Theorem | dfss 3125 |
Variant of subclass definition df-ss 3124. (Contributed by NM,
3-Sep-2004.)
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Theorem | dfss2 3126* |
Alternate definition of the subclass relationship between two classes.
Definition 5.9 of [TakeutiZaring]
p. 17. (Contributed by NM,
8-Jan-2002.)
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Theorem | dfss3 3127* |
Alternate definition of subclass relationship. (Contributed by NM,
14-Oct-1999.)
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Theorem | dfss2f 3128 |
Equivalence for subclass relation, using bound-variable hypotheses
instead of distinct variable conditions. (Contributed by NM,
3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
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Theorem | dfss3f 3129 |
Equivalence for subclass relation, using bound-variable hypotheses
instead of distinct variable conditions. (Contributed by NM,
20-Mar-2004.)
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Theorem | nfss 3130 |
If is not free in and , it is not free in .
(Contributed by NM, 27-Dec-1996.)
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Theorem | ssel 3131 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 5-Aug-1993.)
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Theorem | ssel2 3132 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 7-Jun-2004.)
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Theorem | sseli 3133 |
Membership inference from subclass relationship. (Contributed by NM,
5-Aug-1993.)
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Theorem | sselii 3134 |
Membership inference from subclass relationship. (Contributed by NM,
31-May-1999.)
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Theorem | sseldi 3135 |
Membership inference from subclass relationship. (Contributed by NM,
25-Jun-2014.)
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Theorem | sseld 3136 |
Membership deduction from subclass relationship. (Contributed by NM,
15-Nov-1995.)
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Theorem | sselda 3137 |
Membership deduction from subclass relationship. (Contributed by NM,
26-Jun-2014.)
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Theorem | sseldd 3138 |
Membership inference from subclass relationship. (Contributed by NM,
14-Dec-2004.)
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Theorem | ssneld 3139 |
If a class is not in another class, it is also not in a subclass of that
class. Deduction form. (Contributed by David Moews, 1-May-2017.)
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Theorem | ssneldd 3140 |
If an element is not in a class, it is also not in a subclass of that
class. Deduction form. (Contributed by David Moews, 1-May-2017.)
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Theorem | ssriv 3141* |
Inference based on subclass definition. (Contributed by NM,
5-Aug-1993.)
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Theorem | ssrd 3142 |
Deduction based on subclass definition. (Contributed by Thierry Arnoux,
8-Mar-2017.)
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Theorem | ssrdv 3143* |
Deduction based on subclass definition. (Contributed by NM,
15-Nov-1995.)
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Theorem | sstr2 3144 |
Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17.
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
14-Jun-2011.)
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Theorem | sstr 3145 |
Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by
NM, 5-Sep-2003.)
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Theorem | sstri 3146 |
Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
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Theorem | sstrd 3147 |
Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
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Theorem | sstrid 3148 |
Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
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Theorem | sstrdi 3149 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | sylan9ss 3150 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(Proof shortened by Andrew Salmon, 14-Jun-2011.)
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Theorem | sylan9ssr 3151 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
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Theorem | eqss 3152 |
The subclass relationship is antisymmetric. Compare Theorem 4 of
[Suppes] p. 22. (Contributed by NM,
5-Aug-1993.)
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Theorem | eqssi 3153 |
Infer equality from two subclass relationships. Compare Theorem 4 of
[Suppes] p. 22. (Contributed by NM,
9-Sep-1993.)
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Theorem | eqssd 3154 |
Equality deduction from two subclass relationships. Compare Theorem 4
of [Suppes] p. 22. (Contributed by NM,
27-Jun-2004.)
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Theorem | eqrd 3155 |
Deduce equality of classes from equivalence of membership. (Contributed
by Thierry Arnoux, 21-Mar-2017.)
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Theorem | eqelssd 3156* |
Equality deduction from subclass relationship and membership.
(Contributed by AV, 21-Aug-2022.)
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Theorem | ssid 3157 |
Any class is a subclass of itself. Exercise 10 of [TakeutiZaring]
p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew
Salmon, 14-Jun-2011.)
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Theorem | ssidd 3158 |
Weakening of ssid 3157. (Contributed by BJ, 1-Sep-2022.)
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Theorem | ssv 3159 |
Any class is a subclass of the universal class. (Contributed by NM,
31-Oct-1995.)
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Theorem | sseq1 3160 |
Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof
shortened by Andrew Salmon, 21-Jun-2011.)
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Theorem | sseq2 3161 |
Equality theorem for the subclass relationship. (Contributed by NM,
25-Jun-1998.)
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Theorem | sseq12 3162 |
Equality theorem for the subclass relationship. (Contributed by NM,
31-May-1999.)
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Theorem | sseq1i 3163 |
An equality inference for the subclass relationship. (Contributed by
NM, 18-Aug-1993.)
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Theorem | sseq2i 3164 |
An equality inference for the subclass relationship. (Contributed by
NM, 30-Aug-1993.)
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Theorem | sseq12i 3165 |
An equality inference for the subclass relationship. (Contributed by
NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
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Theorem | sseq1d 3166 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
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Theorem | sseq2d 3167 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
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Theorem | sseq12d 3168 |
An equality deduction for the subclass relationship. (Contributed by
NM, 31-May-1999.)
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Theorem | eqsstri 3169 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 16-Jul-1995.)
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Theorem | eqsstrri 3170 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 19-Oct-1999.)
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Theorem | sseqtri 3171 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 28-Jul-1995.)
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Theorem | sseqtrri 3172 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 4-Apr-1995.)
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Theorem | eqsstrd 3173 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | eqsstrrd 3174 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtrd 3175 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtrrd 3176 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | 3sstr3i 3177 |
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 3178 |
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 3179 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4g 3180 |
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 3181 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4d 3182 |
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 3183 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | eqsstrrid 3184 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | sseqtrdi 3185 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | sseqtrrdi 3186 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | sseqtrid 3187 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | sseqtrrid 3188 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | eqsstrdi 3189 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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Theorem | eqsstrrdi 3190 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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Theorem | eqimss 3191 |
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 3192 |
Equality implies the subclass relation. (Contributed by NM,
23-Nov-2003.)
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Theorem | eqimssi 3193 |
Infer subclass relationship from equality. (Contributed by NM,
6-Jan-2007.)
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Theorem | eqimss2i 3194 |
Infer subclass relationship from equality. (Contributed by NM,
7-Jan-2007.)
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Theorem | nssne1 3195 |
Two classes are different if they don't include the same class.
(Contributed by NM, 23-Apr-2015.)
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Theorem | nssne2 3196 |
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 3197* |
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 3198 |
Demonstrate by witnesses that two classes lack a subclass relation.
(Contributed by Stefan O'Rear, 5-Feb-2015.)
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Theorem | ssrexf 3199 |
Restricted existential quantification follows from a subclass
relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
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Theorem | ssrmof 3200 |
"At most one" existential quantification restricted to a subclass.
(Contributed by Thierry Arnoux, 8-Oct-2017.)
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