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Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeldifd 2901 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 2900. (Contributed by David Moews, 1-May-2017.)
(φA B)    &   (φ → ¬ A 𝐶)       (φA (B𝐶))
 
Theoremeldifad 2902 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 2900. (Contributed by David Moews, 1-May-2017.)
(φA (B𝐶))       (φA B)
 
Theoremeldifbd 2903 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2900. (Contributed by David Moews, 1-May-2017.)
(φA (B𝐶))       (φ → ¬ A 𝐶)
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 2904 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that AA (proved in ssid 2937). Contrast this relationship with the relationship AB (as will be defined in df-pss 2906). For a more traditional definition, but requiring a dummy variable, see dfss2 2907 (or dfss3 2908 which is similar). (Contributed by NM, 27-Apr-1994.)
(AB ↔ (AB) = A)
 
Theoremdfss 2905 Variant of subclass definition df-ss 2904. (Contributed by NM, 3-Sep-2004.)
(ABA = (AB))
 
Definitiondf-pss 2906 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Note that ¬ AA (proved in pssirr 3017). Contrast this relationship with the relationship AB (as defined in df-ss 2904). Other possible definitions are given by dfpss2 3002 and dfpss3 3003. (Contributed by NM, 7-Feb-1996.)
(AB ↔ (AB AB))
 
Theoremdfss2 2907* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
(ABx(x Ax B))
 
Theoremdfss3 2908* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
(ABx A x B)
 
Theoremdfss2f 2909 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.)
xA    &   xB       (ABx(x Ax B))
 
Theoremdfss3f 2910 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
xA    &   xB       (ABx A x B)
 
Theoremnfss 2911 If x is not free in A and B, it is not free in AB. (Contributed by NM, 27-Dec-1996.)
xA    &   xB       x AB
 
Theoremssel 2912 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
(AB → (𝐶 A𝐶 B))
 
Theoremssel2 2913 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
((AB 𝐶 A) → 𝐶 B)
 
Theoremsseli 2914 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
AB       (𝐶 A𝐶 B)
 
Theoremsselii 2915 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
AB    &   𝐶 A       𝐶 B
 
Theoremsseldi 2916 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
AB    &   (φ𝐶 A)       (φ𝐶 B)
 
Theoremsseld 2917 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
(φAB)       (φ → (𝐶 A𝐶 B))
 
Theoremsselda 2918 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
(φAB)       ((φ 𝐶 A) → 𝐶 B)
 
Theoremsseldd 2919 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
(φAB)    &   (φ𝐶 A)       (φ𝐶 B)
 
Theoremssneld 2920 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.)
(φAB)       (φ → (¬ 𝐶 B → ¬ 𝐶 A))
 
Theoremssneldd 2921 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.)
(φAB)    &   (φ → ¬ 𝐶 B)       (φ → ¬ 𝐶 A)
 
Theoremssriv 2922* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)
(x Ax B)       AB
 
Theoremssrd 2923 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
xφ    &   xA    &   xB    &   (φ → (x Ax B))       (φAB)
 
Theoremssrdv 2924* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
(φ → (x Ax B))       (φAB)
 
Theoremsstr2 2925 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(AB → (B𝐶A𝐶))
 
Theoremsstr 2926 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
((AB B𝐶) → A𝐶)
 
Theoremsstri 2927 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
AB    &   B𝐶       A𝐶
 
Theoremsstrd 2928 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
(φAB)    &   (φB𝐶)       (φA𝐶)
 
Theoremsyl5ss 2929 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
AB    &   (φB𝐶)       (φA𝐶)
 
Theoremsyl6ss 2930 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φAB)    &   B𝐶       (φA𝐶)
 
Theoremsylan9ss 2931 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(φAB)    &   (ψB𝐶)       ((φ ψ) → A𝐶)
 
Theoremsylan9ssr 2932 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(φAB)    &   (ψB𝐶)       ((ψ φ) → A𝐶)
 
Theoremeqss 2933 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
(A = B ↔ (AB BA))
 
Theoremeqssi 2934 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
AB    &   BA       A = B
 
Theoremeqssd 2935 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
(φAB)    &   (φBA)       (φA = B)
 
Theoremeqrd 2936 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
xφ    &   xA    &   xB    &   (φ → (x Ax B))       (φA = B)
 
Theoremssid 2937 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.)
AA
 
Theoremssv 2938 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
A ⊆ V
 
Theoremsseq1 2939 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(A = B → (A𝐶B𝐶))
 
Theoremsseq2 2940 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
(A = B → (𝐶A𝐶B))
 
Theoremsseq12 2941 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
((A = B 𝐶 = 𝐷) → (A𝐶B𝐷))
 
Theoremsseq1i 2942 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
A = B       (A𝐶B𝐶)
 
Theoremsseq2i 2943 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
A = B       (𝐶A𝐶B)
 
Theoremsseq12i 2944 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
A = B    &   𝐶 = 𝐷       (A𝐶B𝐷)
 
Theoremsseq1d 2945 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(φA = B)       (φ → (A𝐶B𝐶))
 
Theoremsseq2d 2946 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(φA = B)       (φ → (𝐶A𝐶B))
 
Theoremsseq12d 2947 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶B𝐷))
 
Theoremeqsstri 2948 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
A = B    &   B𝐶       A𝐶
 
Theoremeqsstr3i 2949 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
B = A    &   B𝐶       A𝐶
 
Theoremsseqtri 2950 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
AB    &   B = 𝐶       A𝐶
 
Theoremsseqtr4i 2951 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
AB    &   𝐶 = B       A𝐶
 
Theoremeqsstrd 2952 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φA = B)    &   (φB𝐶)       (φA𝐶)
 
Theoremeqsstr3d 2953 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φB = A)    &   (φB𝐶)       (φA𝐶)
 
Theoremsseqtrd 2954 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φAB)    &   (φB = 𝐶)       (φA𝐶)
 
Theoremsseqtr4d 2955 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φAB)    &   (φ𝐶 = B)       (φA𝐶)
 
Theorem3sstr3i 2956 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
AB    &   A = 𝐶    &   B = 𝐷       𝐶𝐷
 
Theorem3sstr4i 2957 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
AB    &   𝐶 = A    &   𝐷 = B       𝐶𝐷
 
Theorem3sstr3g 2958 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(φAB)    &   A = 𝐶    &   B = 𝐷       (φ𝐶𝐷)
 
Theorem3sstr4g 2959 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(φAB)    &   𝐶 = A    &   𝐷 = B       (φ𝐶𝐷)
 
Theorem3sstr3d 2960 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(φAB)    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐶𝐷)
 
Theorem3sstr4d 2961 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(φAB)    &   (φ𝐶 = A)    &   (φ𝐷 = B)       (φ𝐶𝐷)
 
Theoremsyl5eqss 2962 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
A = B    &   (φB𝐶)       (φA𝐶)
 
Theoremsyl5eqssr 2963 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
B = A    &   (φB𝐶)       (φA𝐶)
 
Theoremsyl6sseq 2964 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(φAB)    &   B = 𝐶       (φA𝐶)
 
Theoremsyl6sseqr 2965 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(φAB)    &   𝐶 = B       (φA𝐶)
 
Theoremsyl5sseq 2966 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
BA    &   (φA = 𝐶)       (φB𝐶)
 
Theoremsyl5sseqr 2967 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
BA    &   (φ𝐶 = A)       (φB𝐶)
 
Theoremsyl6eqss 2968 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(φA = B)    &   B𝐶       (φA𝐶)
 
Theoremsyl6eqssr 2969 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(φB = A)    &   B𝐶       (φA𝐶)
 
Theoremeqimss 2970 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(A = BAB)
 
Theoremeqimss2 2971 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
(B = AAB)
 
Theoremeqimssi 2972 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
A = B       AB
 
Theoremeqimss2i 2973 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
A = B       BA
 
Theoremnssne1 2974 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
((AB ¬ A𝐶) → B𝐶)
 
Theoremnssne2 2975 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
((A𝐶 ¬ B𝐶) → AB)
 
Theoremnssr 2976* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
(x(x A ¬ x B) → ¬ AB)
 
Theoremssralv 2977* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
(AB → (x B φx A φ))
 
Theoremssrexv 2978* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
(AB → (x A φx B φ))
 
Theoremralss 2979* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(AB → (x A φx B (x Aφ)))
 
Theoremrexss 2980* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(AB → (x A φx B (x A φ)))
 
Theoremss2ab 2981 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
({xφ} ⊆ {xψ} ↔ x(φψ))
 
Theoremabss 2982* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({xφ} ⊆ Ax(φx A))
 
Theoremssab 2983* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
(A ⊆ {xφ} ↔ x(x Aφ))
 
Theoremssabral 2984* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
(A ⊆ {xφ} ↔ x A φ)
 
Theoremss2abi 2985 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
(φψ)       {xφ} ⊆ {xψ}
 
Theoremss2abdv 2986* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
(φ → (ψχ))       (φ → {xψ} ⊆ {xχ})
 
Theoremabssdv 2987* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(φ → (ψx A))       (φ → {xψ} ⊆ A)
 
Theoremabssi 2988* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(φx A)       {xφ} ⊆ A
 
Theoremss2rab 2989 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
({x Aφ} ⊆ {x Aψ} ↔ x A (φψ))
 
Theoremrabss 2990* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({x Aφ} ⊆ Bx A (φx B))
 
Theoremssrab 2991* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
(B ⊆ {x Aφ} ↔ (BA x B φ))
 
Theoremssrabdv 2992* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
(φBA)    &   ((φ x B) → ψ)       (φB ⊆ {x Aψ})
 
Theoremrabssdv 2993* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
((φ x A ψ) → x B)       (φ → {x Aψ} ⊆ B)
 
Theoremss2rabdv 2994* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
((φ x A) → (ψχ))       (φ → {x Aψ} ⊆ {x Aχ})
 
Theoremss2rabi 2995 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
(x A → (φψ))       {x Aφ} ⊆ {x Aψ}
 
Theoremrabss2 2996* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(AB → {x Aφ} ⊆ {x Bφ})
 
Theoremssab2 2997* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
{x ∣ (x A φ)} ⊆ A
 
Theoremssrab2 2998* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
{x Aφ} ⊆ A
 
Theoremssrabeq 2999* 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.)
(𝑉 ⊆ {x 𝑉φ} ↔ 𝑉 = {x 𝑉φ})
 
Theoremrabssab 3000 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
{x Aφ} ⊆ {xφ}
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