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Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcsbabg 2901* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(A 𝑉A / x{yφ} = {y[A / x]φ})
 
Theoremcbvralcsf 2902 A more general version of cbvralf 2521 that doesn't require A and B to be distinct from x or y. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
yA    &   xB    &   yφ    &   xψ    &   (x = yA = B)    &   (x = y → (φψ))       (x A φy B ψ)
 
Theoremcbvrexcsf 2903 A more general version of cbvrexf 2522 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.)
yA    &   xB    &   yφ    &   xψ    &   (x = yA = B)    &   (x = y → (φψ))       (x A φy B ψ)
 
Theoremcbvreucsf 2904 A more general version of cbvreuv 2529 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
yA    &   xB    &   yφ    &   xψ    &   (x = yA = B)    &   (x = y → (φψ))       (∃!x A φ∃!y B ψ)
 
Theoremcbvrabcsf 2905 A more general version of cbvrab 2549 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
yA    &   xB    &   yφ    &   xψ    &   (x = yA = B)    &   (x = y → (φψ))       {x Aφ} = {y Bψ}
 
Theoremcbvralv2 2906* 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.)
(x = y → (ψχ))    &   (x = yA = B)       (x A ψy B χ)
 
Theoremcbvrexv2 2907* 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.)
(x = y → (ψχ))    &   (x = yA = B)       (x A ψy B χ)
 
2.1.11  Define basic set operations and relations
 
Syntaxcdif 2908 Extend class notation to include class difference (read: "A minus B").
class (AB)
 
Syntaxcun 2909 Extend class notation to include union of two classes (read: "A union B").
class (AB)
 
Syntaxcin 2910 Extend class notation to include the intersection of two classes (read: "A intersect B").
class (AB)
 
Syntaxwss 2911 Extend wff notation to include the subclass relation. This is read "A is a subclass of B " or "B includes A." When A exists as a set, it is also read "A is a subset of B."
wff AB
 
Syntaxwpss 2912 Extend wff notation with proper subclass relation.
wff AB
 
Theoremdifjust 2913* Soundness justification theorem for df-dif 2914. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{x ∣ (x A ¬ x B)} = {y ∣ (y A ¬ y B)}
 
Definitiondf-dif 2914* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union (AB) (df-un 2916) and intersection (AB) (df-in 2918). 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 "A excludes B " to mean AB. We will use "B is removed from A " to mean A ∖ {B} i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)
(AB) = {x ∣ (x A ¬ x B)}
 
Theoremunjust 2915* Soundness justification theorem for df-un 2916. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{x ∣ (x A x B)} = {y ∣ (y A y B)}
 
Definitiondf-un 2916* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference (AB) (df-dif 2914) and intersection (AB) (df-in 2918). (Contributed by NM, 23-Aug-1993.)
(AB) = {x ∣ (x A x B)}
 
Theoreminjust 2917* Soundness justification theorem for df-in 2918. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{x ∣ (x A x B)} = {y ∣ (y A y B)}
 
Definitiondf-in 2918* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union (AB) (df-un 2916) and difference (AB) (df-dif 2914). (Contributed by NM, 29-Apr-1994.)
(AB) = {x ∣ (x A x B)}
 
Theoremdfin5 2919* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
(AB) = {x Ax B}
 
Theoremdfdif2 2920* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
(AB) = {x A ∣ ¬ x B}
 
Theoremeldif 2921 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(A (B𝐶) ↔ (A B ¬ A 𝐶))
 
Theoremeldifd 2922 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 2921. (Contributed by David Moews, 1-May-2017.)
(φA B)    &   (φ → ¬ A 𝐶)       (φA (B𝐶))
 
Theoremeldifad 2923 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 2921. (Contributed by David Moews, 1-May-2017.)
(φA (B𝐶))       (φA B)
 
Theoremeldifbd 2924 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2921. (Contributed by David Moews, 1-May-2017.)
(φA (B𝐶))       (φ → ¬ A 𝐶)
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 2925 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that AA (proved in ssid 2958). Contrast this relationship with the relationship AB (as will be defined in df-pss 2927). For a more traditional definition, but requiring a dummy variable, see dfss2 2928 (or dfss3 2929 which is similar). (Contributed by NM, 27-Apr-1994.)
(AB ↔ (AB) = A)
 
Theoremdfss 2926 Variant of subclass definition df-ss 2925. (Contributed by NM, 3-Sep-2004.)
(ABA = (AB))
 
Definitiondf-pss 2927 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Note that ¬ AA (proved in pssirr 3038). Contrast this relationship with the relationship AB (as defined in df-ss 2925). Other possible definitions are given by dfpss2 3023 and dfpss3 3024. (Contributed by NM, 7-Feb-1996.)
(AB ↔ (AB AB))
 
Theoremdfss2 2928* 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 2929* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
(ABx A x B)
 
Theoremdfss2f 2930 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 2931 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 2932 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 2933 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
(AB → (𝐶 A𝐶 B))
 
Theoremssel2 2934 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
((AB 𝐶 A) → 𝐶 B)
 
Theoremsseli 2935 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
AB       (𝐶 A𝐶 B)
 
Theoremsselii 2936 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
AB    &   𝐶 A       𝐶 B
 
Theoremsseldi 2937 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
AB    &   (φ𝐶 A)       (φ𝐶 B)
 
Theoremsseld 2938 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
(φAB)       (φ → (𝐶 A𝐶 B))
 
Theoremsselda 2939 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
(φAB)       ((φ 𝐶 A) → 𝐶 B)
 
Theoremsseldd 2940 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
(φAB)    &   (φ𝐶 A)       (φ𝐶 B)
 
Theoremssneld 2941 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 2942 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 2943* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)
(x Ax B)       AB
 
Theoremssrd 2944 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
xφ    &   xA    &   xB    &   (φ → (x Ax B))       (φAB)
 
Theoremssrdv 2945* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
(φ → (x Ax B))       (φAB)
 
Theoremsstr2 2946 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 2947 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
((AB B𝐶) → A𝐶)
 
Theoremsstri 2948 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
AB    &   B𝐶       A𝐶
 
Theoremsstrd 2949 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
(φAB)    &   (φB𝐶)       (φA𝐶)
 
Theoremsyl5ss 2950 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
AB    &   (φB𝐶)       (φA𝐶)
 
Theoremsyl6ss 2951 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φAB)    &   B𝐶       (φA𝐶)
 
Theoremsylan9ss 2952 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(φAB)    &   (ψB𝐶)       ((φ ψ) → A𝐶)
 
Theoremsylan9ssr 2953 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(φAB)    &   (ψB𝐶)       ((ψ φ) → A𝐶)
 
Theoremeqss 2954 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
(A = B ↔ (AB BA))
 
Theoremeqssi 2955 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
AB    &   BA       A = B
 
Theoremeqssd 2956 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
(φAB)    &   (φBA)       (φA = B)
 
Theoremeqrd 2957 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 2958 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 2959 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
A ⊆ V
 
Theoremsseq1 2960 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(A = B → (A𝐶B𝐶))
 
Theoremsseq2 2961 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
(A = B → (𝐶A𝐶B))
 
Theoremsseq12 2962 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
((A = B 𝐶 = 𝐷) → (A𝐶B𝐷))
 
Theoremsseq1i 2963 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
A = B       (A𝐶B𝐶)
 
Theoremsseq2i 2964 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
A = B       (𝐶A𝐶B)
 
Theoremsseq12i 2965 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 2966 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(φA = B)       (φ → (A𝐶B𝐶))
 
Theoremsseq2d 2967 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(φA = B)       (φ → (𝐶A𝐶B))
 
Theoremsseq12d 2968 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶B𝐷))
 
Theoremeqsstri 2969 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
A = B    &   B𝐶       A𝐶
 
Theoremeqsstr3i 2970 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
B = A    &   B𝐶       A𝐶
 
Theoremsseqtri 2971 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
AB    &   B = 𝐶       A𝐶
 
Theoremsseqtr4i 2972 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
AB    &   𝐶 = B       A𝐶
 
Theoremeqsstrd 2973 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φA = B)    &   (φB𝐶)       (φA𝐶)
 
Theoremeqsstr3d 2974 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φB = A)    &   (φB𝐶)       (φA𝐶)
 
Theoremsseqtrd 2975 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φAB)    &   (φB = 𝐶)       (φA𝐶)
 
Theoremsseqtr4d 2976 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φAB)    &   (φ𝐶 = B)       (φA𝐶)
 
Theorem3sstr3i 2977 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 2978 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 2979 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(φAB)    &   A = 𝐶    &   B = 𝐷       (φ𝐶𝐷)
 
Theorem3sstr4g 2980 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 2981 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(φAB)    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐶𝐷)
 
Theorem3sstr4d 2982 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 2983 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
A = B    &   (φB𝐶)       (φA𝐶)
 
Theoremsyl5eqssr 2984 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
B = A    &   (φB𝐶)       (φA𝐶)
 
Theoremsyl6sseq 2985 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(φAB)    &   B = 𝐶       (φA𝐶)
 
Theoremsyl6sseqr 2986 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(φAB)    &   𝐶 = B       (φA𝐶)
 
Theoremsyl5sseq 2987 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
BA    &   (φA = 𝐶)       (φB𝐶)
 
Theoremsyl5sseqr 2988 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
BA    &   (φ𝐶 = A)       (φB𝐶)
 
Theoremsyl6eqss 2989 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(φA = B)    &   B𝐶       (φA𝐶)
 
Theoremsyl6eqssr 2990 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(φB = A)    &   B𝐶       (φA𝐶)
 
Theoremeqimss 2991 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(A = BAB)
 
Theoremeqimss2 2992 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
(B = AAB)
 
Theoremeqimssi 2993 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
A = B       AB
 
Theoremeqimss2i 2994 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
A = B       BA
 
Theoremnssne1 2995 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
((AB ¬ A𝐶) → B𝐶)
 
Theoremnssne2 2996 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
((A𝐶 ¬ B𝐶) → AB)
 
Theoremnssr 2997* 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 2998* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
(AB → (x B φx A φ))
 
Theoremssrexv 2999* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
(AB → (x A φx B φ))
 
Theoremralss 3000* 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)))
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