Theorem List for Intuitionistic Logic Explorer - 3001-3100 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | rmob 3001* |
Consequence of "at most one", using implicit substitution.
(Contributed
by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
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Theorem | rmoi 3002* |
Consequence of "at most one", using implicit substitution.
(Contributed
by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
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2.1.10 Proper substitution of classes for sets
into classes
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Syntax | csb 3003 |
Extend class notation to include the proper substitution of a class for a
set into another class.
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Definition | df-csb 3004* |
Define the proper substitution of a class for a set into another class.
The underlined brackets distinguish it from the substitution into a wff,
wsbc 2909, to prevent ambiguity. Theorem sbcel1g 3021 shows an example of
how ambiguity could arise if we didn't use distinguished brackets.
Theorem sbccsbg 3031 recreates substitution into a wff from this
definition. (Contributed by NM, 10-Nov-2005.)
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Theorem | csb2 3005* |
Alternate expression for the proper substitution into a class, without
referencing substitution into a wff. Note that can be free in
but cannot
occur in .
(Contributed by NM, 2-Dec-2013.)
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Theorem | csbeq1 3006 |
Analog of dfsbcq 2911 for proper substitution into a class.
(Contributed
by NM, 10-Nov-2005.)
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Theorem | cbvcsbw 3007* |
Version of cbvcsb 3008 with a disjoint variable condition.
(Contributed by
Gino Giotto, 10-Jan-2024.)
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Theorem | cbvcsb 3008 |
Change bound variables in a class substitution. Interestingly, this
does not require any bound variable conditions on . (Contributed
by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro,
11-Dec-2016.)
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Theorem | cbvcsbv 3009* |
Change the bound variable of a proper substitution into a class using
implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by
Mario Carneiro, 13-Oct-2016.)
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Theorem | csbeq1d 3010 |
Equality deduction for proper substitution into a class. (Contributed
by NM, 3-Dec-2005.)
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Theorem | csbid 3011 |
Analog of sbid 1747 for proper substitution into a class.
(Contributed by
NM, 10-Nov-2005.)
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Theorem | csbeq1a 3012 |
Equality theorem for proper substitution into a class. (Contributed by
NM, 10-Nov-2005.)
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Theorem | csbco 3013* |
Composition law for chained substitutions into a class. (Contributed by
NM, 10-Nov-2005.)
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Theorem | csbtt 3014 |
Substitution doesn't affect a constant (in which is not
free). (Contributed by Mario Carneiro, 14-Oct-2016.)
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Theorem | csbconstgf 3015 |
Substitution doesn't affect a constant (in which is not
free). (Contributed by NM, 10-Nov-2005.)
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Theorem | csbconstg 3016* |
Substitution doesn't affect a constant (in which is not
free). csbconstgf 3015 with distinct variable requirement.
(Contributed by
Alan Sare, 22-Jul-2012.)
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Theorem | sbcel12g 3017 |
Distribute proper substitution through a membership relation.
(Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon,
29-Jun-2011.)
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Theorem | sbceqg 3018 |
Distribute proper substitution through an equality relation.
(Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon,
29-Jun-2011.)
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Theorem | sbcnel12g 3019 |
Distribute proper substitution through negated membership. (Contributed
by Andrew Salmon, 18-Jun-2011.)
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Theorem | sbcne12g 3020 |
Distribute proper substitution through an inequality. (Contributed by
Andrew Salmon, 18-Jun-2011.)
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Theorem | sbcel1g 3021* |
Move proper substitution in and out of a membership relation. Note that
the scope of is the wff , whereas the scope
of is the class . (Contributed by NM,
10-Nov-2005.)
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Theorem | sbceq1g 3022* |
Move proper substitution to first argument of an equality. (Contributed
by NM, 30-Nov-2005.)
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Theorem | sbcel2g 3023* |
Move proper substitution in and out of a membership relation.
(Contributed by NM, 14-Nov-2005.)
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Theorem | sbceq2g 3024* |
Move proper substitution to second argument of an equality.
(Contributed by NM, 30-Nov-2005.)
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Theorem | csbcomg 3025* |
Commutative law for double substitution into a class. (Contributed by
NM, 14-Nov-2005.)
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Theorem | csbeq2 3026 |
Substituting into equivalent classes gives equivalent results.
(Contributed by Giovanni Mascellani, 9-Apr-2018.)
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Theorem | csbeq2d 3027 |
Formula-building deduction for class substitution. (Contributed by NM,
22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
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Theorem | csbeq2dv 3028* |
Formula-building deduction for class substitution. (Contributed by NM,
10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
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Theorem | csbeq2i 3029 |
Formula-building inference for class substitution. (Contributed by NM,
10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
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Theorem | csbvarg 3030 |
The proper substitution of a class for setvar variable results in the
class (if the class exists). (Contributed by NM, 10-Nov-2005.)
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Theorem | sbccsbg 3031* |
Substitution into a wff expressed in terms of substitution into a class.
(Contributed by NM, 15-Aug-2007.)
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Theorem | sbccsb2g 3032 |
Substitution into a wff expressed in using substitution into a class.
(Contributed by NM, 27-Nov-2005.)
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Theorem | nfcsb1d 3033 |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by Mario Carneiro, 12-Oct-2016.)
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Theorem | nfcsb1 3034 |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by Mario Carneiro, 12-Oct-2016.)
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Theorem | nfcsb1v 3035* |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro,
12-Oct-2016.)
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Theorem | nfcsbd 3036 |
Deduction version of nfcsb 3037. (Contributed by NM, 21-Nov-2005.)
(Revised by Mario Carneiro, 12-Oct-2016.)
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Theorem | nfcsb 3037 |
Bound-variable hypothesis builder for substitution into a class.
(Contributed by Mario Carneiro, 12-Oct-2016.)
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Theorem | csbhypf 3038* |
Introduce an explicit substitution into an implicit substitution
hypothesis. See sbhypf 2735 for class substitution version. (Contributed
by NM, 19-Dec-2008.)
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Theorem | csbiebt 3039* |
Conversion of implicit substitution to explicit substitution into a
class. (Closed theorem version of csbiegf 3043.) (Contributed by NM,
11-Nov-2005.)
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Theorem | csbiedf 3040* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by Mario Carneiro, 13-Oct-2016.)
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Theorem | csbieb 3041* |
Bidirectional conversion between an implicit class substitution
hypothesis and its explicit substitution equivalent.
(Contributed by NM, 2-Mar-2008.)
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Theorem | csbiebg 3042* |
Bidirectional conversion between an implicit class substitution
hypothesis and its explicit substitution equivalent.
(Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro,
11-Dec-2016.)
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Theorem | csbiegf 3043* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro,
13-Oct-2016.)
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Theorem | csbief 3044* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro,
13-Oct-2016.)
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Theorem | csbie 3045* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by AV, 2-Dec-2019.)
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Theorem | csbied 3046* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario
Carneiro, 13-Oct-2016.)
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Theorem | csbied2 3047* |
Conversion of implicit substitution to explicit class substitution,
deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
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Theorem | csbie2t 3048* |
Conversion of implicit substitution to explicit substitution into a
class (closed form of csbie2 3049). (Contributed by NM, 3-Sep-2007.)
(Revised by Mario Carneiro, 13-Oct-2016.)
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Theorem | csbie2 3049* |
Conversion of implicit substitution to explicit substitution into a
class. (Contributed by NM, 27-Aug-2007.)
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Theorem | csbie2g 3050* |
Conversion of implicit substitution to explicit class substitution.
This version of sbcie 2943 avoids a disjointness condition on and
by
substituting twice. (Contributed by Mario Carneiro,
11-Nov-2016.)
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Theorem | sbcnestgf 3051 |
Nest the composition of two substitutions. (Contributed by Mario
Carneiro, 11-Nov-2016.)
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Theorem | csbnestgf 3052 |
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 3053* |
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 3054* |
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 3055 |
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 3056* |
Idempotent law for class substitutions. (Contributed by NM,
1-Mar-2008.)
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Theorem | sbcco3g 3057* |
Composition of two substitutions. (Contributed by NM, 27-Nov-2005.)
(Revised by Mario Carneiro, 11-Nov-2016.)
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Theorem | csbco3g 3058* |
Composition of two class substitutions. (Contributed by NM,
27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
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Theorem | rspcsbela 3059* |
Special case related to rspsbc 2991. (Contributed by NM, 10-Dec-2005.)
(Proof shortened by Eric Schmidt, 17-Jan-2007.)
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Theorem | sbnfc2 3060* |
Two ways of expressing " is (effectively) not free in ."
(Contributed by Mario Carneiro, 14-Oct-2016.)
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Theorem | csbabg 3061* |
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 3062 |
A more general version of cbvralf 2648 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 3063 |
A more general version of cbvrexf 2649 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 3064 |
A more general version of cbvreuv 2656 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 3065 |
A more general version of cbvrab 2684 with no distinct variable
restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
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Theorem | cbvralv2 3066* |
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 3067* |
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 3068 |
Extend class notation to include class difference (read: " minus
").
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Syntax | cun 3069 |
Extend class notation to include union of two classes (read: "
union ").
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Syntax | cin 3070 |
Extend class notation to include the intersection of two classes (read:
" intersect
").
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Syntax | wss 3071 |
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 3072* |
Soundness justification theorem for df-dif 3073. (Contributed by Rodolfo
Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Definition | df-dif 3073* |
Define class difference, also called relative complement. Definition
5.12 of [TakeutiZaring] p. 20.
Contrast this operation with union
(df-un 3075) and intersection (df-in 3077).
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 3074* |
Soundness justification theorem for df-un 3075. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Definition | df-un 3075* |
Define the union of two classes. Definition 5.6 of [TakeutiZaring]
p. 16. Contrast this operation with difference
(df-dif 3073) and intersection (df-in 3077). (Contributed
by NM, 23-Aug-1993.)
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Theorem | injust 3076* |
Soundness justification theorem for df-in 3077. (Contributed by Rodolfo
Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
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Definition | df-in 3077* |
Define the intersection of two classes. Definition 5.6 of
[TakeutiZaring] p. 16. Contrast
this operation with union
(df-un 3075) and difference (df-dif 3073).
(Contributed by NM, 29-Apr-1994.)
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Theorem | dfin5 3078* |
Alternate definition for the intersection of two classes. (Contributed
by NM, 6-Jul-2005.)
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Theorem | dfdif2 3079* |
Alternate definition of class difference. (Contributed by NM,
25-Mar-2004.)
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Theorem | eldif 3080 |
Expansion of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
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Theorem | eldifd 3081 |
If a class is in one class and not another, it is also in their
difference. One-way deduction form of eldif 3080. (Contributed by David
Moews, 1-May-2017.)
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Theorem | eldifad 3082 |
If a class is in the difference of two classes, it is also in the
minuend. One-way deduction form of eldif 3080. (Contributed by David
Moews, 1-May-2017.)
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Theorem | eldifbd 3083 |
If a class is in the difference of two classes, it is not in the
subtrahend. One-way deduction form of eldif 3080. (Contributed by David
Moews, 1-May-2017.)
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2.1.12 Subclasses and subsets
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Definition | df-ss 3084 |
Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18.
Note that (proved in ssid 3117). For a more traditional
definition, but requiring a dummy variable, see dfss2 3086. Other possible
definitions are given by dfss3 3087, ssequn1 3246, ssequn2 3249, and sseqin2 3295.
(Contributed by NM, 27-Apr-1994.)
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Theorem | dfss 3085 |
Variant of subclass definition df-ss 3084. (Contributed by NM,
3-Sep-2004.)
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Theorem | dfss2 3086* |
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 3087* |
Alternate definition of subclass relationship. (Contributed by NM,
14-Oct-1999.)
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Theorem | dfss2f 3088 |
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 3089 |
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 3090 |
If is not free in and , it is not free in .
(Contributed by NM, 27-Dec-1996.)
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Theorem | ssel 3091 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 5-Aug-1993.)
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Theorem | ssel2 3092 |
Membership relationships follow from a subclass relationship.
(Contributed by NM, 7-Jun-2004.)
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Theorem | sseli 3093 |
Membership inference from subclass relationship. (Contributed by NM,
5-Aug-1993.)
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Theorem | sselii 3094 |
Membership inference from subclass relationship. (Contributed by NM,
31-May-1999.)
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Theorem | sseldi 3095 |
Membership inference from subclass relationship. (Contributed by NM,
25-Jun-2014.)
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Theorem | sseld 3096 |
Membership deduction from subclass relationship. (Contributed by NM,
15-Nov-1995.)
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Theorem | sselda 3097 |
Membership deduction from subclass relationship. (Contributed by NM,
26-Jun-2014.)
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Theorem | sseldd 3098 |
Membership inference from subclass relationship. (Contributed by NM,
14-Dec-2004.)
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Theorem | ssneld 3099 |
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 3100 |
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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