Theorem List for Intuitionistic Logic Explorer - 3101-3200 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | csbnestgf 3101 |
Nest the composition of two substitutions. (Contributed by NM,
23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
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Theorem | sbcnestg 3102* |
Nest the composition of two substitutions. (Contributed by NM,
27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
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Theorem | csbnestg 3103* |
Nest the composition of two substitutions. (Contributed by NM,
23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
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Theorem | csbnest1g 3104 |
Nest the composition of two substitutions. (Contributed by NM,
23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
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Theorem | csbidmg 3105* |
Idempotent law for class substitutions. (Contributed by NM,
1-Mar-2008.)
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Theorem | sbcco3g 3106* |
Composition of two substitutions. (Contributed by NM, 27-Nov-2005.)
(Revised by Mario Carneiro, 11-Nov-2016.)
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Theorem | csbco3g 3107* |
Composition of two class substitutions. (Contributed by NM,
27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
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Theorem | rspcsbela 3108* |
Special case related to rspsbc 3037. (Contributed by NM, 10-Dec-2005.)
(Proof shortened by Eric Schmidt, 17-Jan-2007.)
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Theorem | sbnfc2 3109* |
Two ways of expressing " is (effectively) not free in ."
(Contributed by Mario Carneiro, 14-Oct-2016.)
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Theorem | csbabg 3110* |
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 3111 |
A more general version of cbvralf 2689 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 3112 |
A more general version of cbvrexf 2690 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 3113 |
A more general version of cbvreuv 2698 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 3114 |
A more general version of cbvrab 2728 with no distinct variable
restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
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Theorem | cbvralv2 3115* |
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 3116* |
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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Theorem | rspc2vd 3117* |
Deduction version of 2-variable restricted specialization, using
implicit substitution. Notice that the class for the second set
variable may
depend on the first set variable .
(Contributed by AV, 29-Mar-2021.)
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2.1.11 Define basic set operations and
relations
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Syntax | cdif 3118 |
Extend class notation to include class difference (read: " minus
").
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Syntax | cun 3119 |
Extend class notation to include union of two classes (read: "
union ").
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Syntax | cin 3120 |
Extend class notation to include the intersection of two classes (read:
" intersect
").
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Syntax | wss 3121 |
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 3122* |
Soundness justification theorem for df-dif 3123. (Contributed by Rodolfo
Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Definition | df-dif 3123* |
Define class difference, also called relative complement. Definition
5.12 of [TakeutiZaring] p. 20.
Contrast this operation with union
(df-un 3125) and intersection (df-in 3127).
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 3124* |
Soundness justification theorem for df-un 3125. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Definition | df-un 3125* |
Define the union of two classes. Definition 5.6 of [TakeutiZaring]
p. 16. Contrast this operation with difference
(df-dif 3123) and intersection (df-in 3127). (Contributed
by NM, 23-Aug-1993.)
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Theorem | injust 3126* |
Soundness justification theorem for df-in 3127. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Definition | df-in 3127* |
Define the intersection of two classes. Definition 5.6 of
[TakeutiZaring] p. 16. Contrast
this operation with union
(df-un 3125) and difference (df-dif 3123).
(Contributed by NM, 29-Apr-1994.)
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Theorem | dfin5 3128* |
Alternate definition for the intersection of two classes. (Contributed
by NM, 6-Jul-2005.)
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Theorem | dfdif2 3129* |
Alternate definition of class difference. (Contributed by NM,
25-Mar-2004.)
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Theorem | eldif 3130 |
Expansion of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
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Theorem | eldifd 3131 |
If a class is in one class and not another, it is also in their
difference. One-way deduction form of eldif 3130. (Contributed by David
Moews, 1-May-2017.)
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Theorem | eldifad 3132 |
If a class is in the difference of two classes, it is also in the
minuend. One-way deduction form of eldif 3130. (Contributed by David
Moews, 1-May-2017.)
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Theorem | eldifbd 3133 |
If a class is in the difference of two classes, it is not in the
subtrahend. One-way deduction form of eldif 3130. (Contributed by David
Moews, 1-May-2017.)
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2.1.12 Subclasses and subsets
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Definition | df-ss 3134 |
Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18.
Note that (proved in ssid 3167). For a more traditional
definition, but requiring a dummy variable, see dfss2 3136. Other possible
definitions are given by dfss3 3137, ssequn1 3297, ssequn2 3300, and sseqin2 3346.
(Contributed by NM, 27-Apr-1994.)
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Theorem | dfss 3135 |
Variant of subclass definition df-ss 3134. (Contributed by NM,
3-Sep-2004.)
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Theorem | dfss2 3136* |
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 3137* |
Alternate definition of subclass relationship. (Contributed by NM,
14-Oct-1999.)
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Theorem | dfss2f 3138 |
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 3139 |
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 3140 |
If is not free in and , it is not free in .
(Contributed by NM, 27-Dec-1996.)
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Theorem | ssel 3141 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 5-Aug-1993.)
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Theorem | ssel2 3142 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 7-Jun-2004.)
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Theorem | sseli 3143 |
Membership inference from subclass relationship. (Contributed by NM,
5-Aug-1993.)
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Theorem | sselii 3144 |
Membership inference from subclass relationship. (Contributed by NM,
31-May-1999.)
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Theorem | sselid 3145 |
Membership inference from subclass relationship. (Contributed by NM,
25-Jun-2014.)
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Theorem | sseld 3146 |
Membership deduction from subclass relationship. (Contributed by NM,
15-Nov-1995.)
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Theorem | sselda 3147 |
Membership deduction from subclass relationship. (Contributed by NM,
26-Jun-2014.)
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Theorem | sseldd 3148 |
Membership inference from subclass relationship. (Contributed by NM,
14-Dec-2004.)
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Theorem | ssneld 3149 |
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 3150 |
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 3151* |
Inference based on subclass definition. (Contributed by NM,
5-Aug-1993.)
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Theorem | ssrd 3152 |
Deduction based on subclass definition. (Contributed by Thierry Arnoux,
8-Mar-2017.)
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Theorem | ssrdv 3153* |
Deduction based on subclass definition. (Contributed by NM,
15-Nov-1995.)
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Theorem | sstr2 3154 |
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 3155 |
Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by
NM, 5-Sep-2003.)
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Theorem | sstri 3156 |
Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
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Theorem | sstrd 3157 |
Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
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Theorem | sstrid 3158 |
Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
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Theorem | sstrdi 3159 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | sylan9ss 3160 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(Proof shortened by Andrew Salmon, 14-Jun-2011.)
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Theorem | sylan9ssr 3161 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
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Theorem | eqss 3162 |
The subclass relationship is antisymmetric. Compare Theorem 4 of
[Suppes] p. 22. (Contributed by NM,
5-Aug-1993.)
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Theorem | eqssi 3163 |
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 3164 |
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 3165 |
Deduce equality of classes from equivalence of membership. (Contributed
by Thierry Arnoux, 21-Mar-2017.)
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Theorem | eqelssd 3166* |
Equality deduction from subclass relationship and membership.
(Contributed by AV, 21-Aug-2022.)
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Theorem | ssid 3167 |
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 3168 |
Weakening of ssid 3167. (Contributed by BJ, 1-Sep-2022.)
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Theorem | ssv 3169 |
Any class is a subclass of the universal class. (Contributed by NM,
31-Oct-1995.)
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Theorem | sseq1 3170 |
Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof
shortened by Andrew Salmon, 21-Jun-2011.)
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Theorem | sseq2 3171 |
Equality theorem for the subclass relationship. (Contributed by NM,
25-Jun-1998.)
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Theorem | sseq12 3172 |
Equality theorem for the subclass relationship. (Contributed by NM,
31-May-1999.)
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Theorem | sseq1i 3173 |
An equality inference for the subclass relationship. (Contributed by
NM, 18-Aug-1993.)
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Theorem | sseq2i 3174 |
An equality inference for the subclass relationship. (Contributed by
NM, 30-Aug-1993.)
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Theorem | sseq12i 3175 |
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 3176 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
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Theorem | sseq2d 3177 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
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Theorem | sseq12d 3178 |
An equality deduction for the subclass relationship. (Contributed by
NM, 31-May-1999.)
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Theorem | eqsstri 3179 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 16-Jul-1995.)
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Theorem | eqsstrri 3180 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 19-Oct-1999.)
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Theorem | sseqtri 3181 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 28-Jul-1995.)
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Theorem | sseqtrri 3182 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 4-Apr-1995.)
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Theorem | eqsstrd 3183 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | eqsstrrd 3184 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtrd 3185 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtrrd 3186 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | 3sstr3i 3187 |
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 3188 |
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 3189 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4g 3190 |
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 3191 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4d 3192 |
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 3193 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | eqsstrrid 3194 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | sseqtrdi 3195 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | sseqtrrdi 3196 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | sseqtrid 3197 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | sseqtrrid 3198 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | eqsstrdi 3199 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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Theorem | eqsstrrdi 3200 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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