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Theorem List for Intuitionistic Logic Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsseq2d 3001 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Theoremsseq12d 3002 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremeqsstri 3003 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)

Theoremeqsstr3i 3004 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)

Theoremsseqtri 3005 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)

Theoremsseqtr4i 3006 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)

Theoremeqsstrd 3007 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremeqsstr3d 3008 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrd 3009 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtr4d 3010 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theorem3sstr3i 3011 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theorem3sstr4i 3012 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theorem3sstr3g 3013 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4g 3014 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theorem3sstr3d 3015 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4d 3016 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremsyl5eqss 3017 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl5eqssr 3018 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseq 3019 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseqr 3020 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl5sseq 3021 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsyl5sseqr 3022 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsyl6eqss 3023 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremsyl6eqssr 3024 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremeqimss 3025 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremeqimss2 3026 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)

Theoremeqimssi 3027 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)

Theoremeqimss2i 3028 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)

Theoremnssne1 3029 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnssne2 3030 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnssr 3031* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)

Theoremssralv 3032* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)

Theoremssrexv 3033* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)

Theoremralss 3034* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremrexss 3035* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremss2ab 3036 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)

Theoremabss 3037* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssab 3038* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssabral 3039* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)

Theoremss2abi 3040 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)

Theoremss2abdv 3041* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)

Theoremabssdv 3042* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremabssi 3043* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremss2rab 3044 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)

Theoremrabss 3045* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssrab 3046* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssrabdv 3047* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)

Theoremrabssdv 3048* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)

Theoremss2rabdv 3049* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)

Theoremss2rabi 3050 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)

Theoremrabss2 3051* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssab2 3052* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)

Theoremssrab2 3053* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)

Theoremssrabeq 3054* 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.)

Theoremrabssab 3055 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremuniiunlem 3056* 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.)

Theoremdfpss2 3057 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)

Theoremdfpss3 3058 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempsseq1 3059 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)

Theorempsseq2 3060 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)

Theorempsseq1i 3061 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq2i 3062 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq12i 3063 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq1d 3064 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq2d 3065 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq12d 3066 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempssss 3067 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssne 3068 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)

Theorempssssd 3069 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)

Theorempssned 3070 Proper subclasses are unequal. Deduction form of pssne 3068. (Contributed by David Moews, 1-May-2017.)

Theoremsspssr 3071 Subclass in terms of proper subclass. (Contributed by Jim Kingdon, 16-Jul-2018.)

Theorempssirr 3072 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssn2lp 3073 Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremsspsstrir 3074 Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)

Theoremssnpss 3075 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremsspssn 3076 Like pssn2lp 3073 but for subset and proper subset. (Contributed by Jim Kingdon, 17-Jul-2018.)

Theorempsstr 3077 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theoremsspsstr 3078 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Theorempsssstr 3079 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Theorempsstrd 3080 Proper subclass inclusion is transitive. Deduction form of psstr 3077. (Contributed by David Moews, 1-May-2017.)

Theoremsspsstrd 3081 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3078. (Contributed by David Moews, 1-May-2017.)

Theorempsssstrd 3082 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3079. (Contributed by David Moews, 1-May-2017.)

2.1.13  The difference, union, and intersection of two classes

2.1.13.1  The difference of two classes

Theoremdifeq1 3083 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq2 3084 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq12 3085 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)

Theoremdifeq1i 3086 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2i 3087 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12i 3088 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)

Theoremdifeq1d 3089 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2d 3090 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12d 3091 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)

Theoremdifeqri 3092* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnfdif 3093 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremeldifi 3094 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)

Theoremeldifn 3095 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)

Theoremelndif 3096 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)

Theoremdifdif 3097 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)

Theoremdifss 3098 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)

Theoremdifssd 3099 A difference of two classes is contained in the minuend. Deduction form of difss 3098. (Contributed by David Moews, 1-May-2017.)

Theoremdifss2 3100 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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