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Theorem List for Intuitionistic Logic Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssab2 3001* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
{x ∣ (x A φ)} ⊆ A
 
Theoremssrab2 3002* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
{x Aφ} ⊆ A
 
Theoremssrabeq 3003* 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 3004 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
{x Aφ} ⊆ {xφ}
 
Theoremuniiunlem 3005* 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.)
(x A B 𝐷 → (x A B 𝐶 ↔ {yx A y = B} ⊆ 𝐶))
 
Theoremdfpss2 3006 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
(AB ↔ (AB ¬ A = B))
 
Theoremdfpss3 3007 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(AB ↔ (AB ¬ BA))
 
Theorempsseq1 3008 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(A = B → (A𝐶B𝐶))
 
Theorempsseq2 3009 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(A = B → (𝐶A𝐶B))
 
Theorempsseq1i 3010 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
A = B       (A𝐶B𝐶)
 
Theorempsseq2i 3011 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
A = B       (𝐶A𝐶B)
 
Theorempsseq12i 3012 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
A = B    &   𝐶 = 𝐷       (A𝐶B𝐷)
 
Theorempsseq1d 3013 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(φA = B)       (φ → (A𝐶B𝐶))
 
Theorempsseq2d 3014 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(φA = B)       (φ → (𝐶A𝐶B))
 
Theorempsseq12d 3015 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶B𝐷))
 
Theorempssss 3016 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
(ABAB)
 
Theorempssne 3017 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
(ABAB)
 
Theorempssssd 3018 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
(φAB)       (φAB)
 
Theorempssned 3019 Proper subclasses are unequal. Deduction form of pssne 3017. (Contributed by David Moews, 1-May-2017.)
(φAB)       (φAB)
 
Theoremsspssr 3020 Subclass in terms of proper subclass. (Contributed by Jim Kingdon, 16-Jul-2018.)
((AB A = B) → AB)
 
Theorempssirr 3021 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
¬ AA
 
Theorempssn2lp 3022 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.)
¬ (AB BA)
 
Theoremsspsstrir 3023 Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)
((AB A = B BA) → (AB BA))
 
Theoremssnpss 3024 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(AB → ¬ BA)
 
Theoremsspssn 3025 Like pssn2lp 3022 but for subset and proper subset. (Contributed by Jim Kingdon, 17-Jul-2018.)
¬ (AB BA)
 
Theorempsstr 3026 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
((AB B𝐶) → A𝐶)
 
Theoremsspsstr 3027 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((AB B𝐶) → A𝐶)
 
Theorempsssstr 3028 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((AB B𝐶) → A𝐶)
 
Theorempsstrd 3029 Proper subclass inclusion is transitive. Deduction form of psstr 3026. (Contributed by David Moews, 1-May-2017.)
(φAB)    &   (φB𝐶)       (φA𝐶)
 
Theoremsspsstrd 3030 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3027. (Contributed by David Moews, 1-May-2017.)
(φAB)    &   (φB𝐶)       (φA𝐶)
 
Theorempsssstrd 3031 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3028. (Contributed by David Moews, 1-May-2017.)
(φAB)    &   (φB𝐶)       (φA𝐶)
 
2.1.13  The difference, union, and intersection of two classes
 
2.1.13.1  The difference of two classes
 
Theoremdifeq1 3032 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(A = B → (A𝐶) = (B𝐶))
 
Theoremdifeq2 3033 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(A = B → (𝐶A) = (𝐶B))
 
Theoremdifeq12 3034 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
((A = B 𝐶 = 𝐷) → (A𝐶) = (B𝐷))
 
Theoremdifeq1i 3035 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
A = B       (A𝐶) = (B𝐶)
 
Theoremdifeq2i 3036 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
A = B       (𝐶A) = (𝐶B)
 
Theoremdifeq12i 3037 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
A = B    &   𝐶 = 𝐷       (A𝐶) = (B𝐷)
 
Theoremdifeq1d 3038 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
(φA = B)       (φ → (A𝐶) = (B𝐶))
 
Theoremdifeq2d 3039 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
(φA = B)       (φ → (𝐶A) = (𝐶B))
 
Theoremdifeq12d 3040 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))
 
Theoremdifeqri 3041* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((x A ¬ x B) ↔ x 𝐶)       (AB) = 𝐶
 
Theoremnfdif 3042 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
xA    &   xB       x(AB)
 
Theoremeldifi 3043 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(A (B𝐶) → A B)
 
Theoremeldifn 3044 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
(A (B𝐶) → ¬ A 𝐶)
 
Theoremelndif 3045 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
(A B → ¬ A (𝐶B))
 
Theoremdifdif 3046 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
(A ∖ (BA)) = A
 
Theoremdifss 3047 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(AB) ⊆ A
 
Theoremdifssd 3048 A difference of two classes is contained in the minuend. Deduction form of difss 3047. (Contributed by David Moews, 1-May-2017.)
(φ → (AB) ⊆ A)
 
Theoremdifss2 3049 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
(A ⊆ (B𝐶) → AB)
 
Theoremdifss2d 3050 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3049. (Contributed by David Moews, 1-May-2017.)
(φA ⊆ (B𝐶))       (φAB)
 
Theoremssdifss 3051 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
(AB → (A𝐶) ⊆ B)
 
Theoremddifnel 3052* Double complement under universal class. The hypothesis is one way of expressing the idea that membership in A is decidable. Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that A is a subset of V ∖ (V ∖ A), see ddifss 3152. (Contributed by Jim Kingdon, 21-Jul-2018.)
x (V ∖ A) → x A)       (V ∖ (V ∖ A)) = A
 
Theoremssconb 3053 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
((A𝐶 B𝐶) → (A ⊆ (𝐶B) ↔ B ⊆ (𝐶A)))
 
Theoremsscon 3054 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
(AB → (𝐶B) ⊆ (𝐶A))
 
Theoremssdif 3055 Difference law for subsets. (Contributed by NM, 28-May-1998.)
(AB → (A𝐶) ⊆ (B𝐶))
 
Theoremssdifd 3056 If A is contained in B, then (A𝐶) is contained in (B𝐶). Deduction form of ssdif 3055. (Contributed by David Moews, 1-May-2017.)
(φAB)       (φ → (A𝐶) ⊆ (B𝐶))
 
Theoremsscond 3057 If A is contained in B, then (𝐶B) is contained in (𝐶A). Deduction form of sscon 3054. (Contributed by David Moews, 1-May-2017.)
(φAB)       (φ → (𝐶B) ⊆ (𝐶A))
 
Theoremssdifssd 3058 If A is contained in B, then (A𝐶) is also contained in B. Deduction form of ssdifss 3051. (Contributed by David Moews, 1-May-2017.)
(φAB)       (φ → (A𝐶) ⊆ B)
 
Theoremssdif2d 3059 If A is contained in B and 𝐶 is contained in 𝐷, then (A𝐷) is contained in (B𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.)
(φAB)    &   (φ𝐶𝐷)       (φ → (A𝐷) ⊆ (B𝐶))
 
Theoremraldifb 3060 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
(x A (xBφ) ↔ x (AB)φ)
 
2.1.13.2  The union of two classes
 
Theoremelun 3061 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
(A (B𝐶) ↔ (A B A 𝐶))
 
Theoremuneqri 3062* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
((x A x B) ↔ x 𝐶)       (AB) = 𝐶
 
Theoremunidm 3063 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
(AA) = A
 
Theoremuncom 3064 Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(AB) = (BA)
 
Theoremequncom 3065 If a class equals the union of two other classes, then it equals the union of those two classes commuted. (Contributed by Alan Sare, 18-Feb-2012.)
(A = (B𝐶) ↔ A = (𝐶B))
 
Theoremequncomi 3066 Inference form of equncom 3065. (Contributed by Alan Sare, 18-Feb-2012.)
A = (B𝐶)       A = (𝐶B)
 
Theoremuneq1 3067 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
(A = B → (A𝐶) = (B𝐶))
 
Theoremuneq2 3068 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
(A = B → (𝐶A) = (𝐶B))
 
Theoremuneq12 3069 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
((A = B 𝐶 = 𝐷) → (A𝐶) = (B𝐷))
 
Theoremuneq1i 3070 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
A = B       (A𝐶) = (B𝐶)
 
Theoremuneq2i 3071 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
A = B       (𝐶A) = (𝐶B)
 
Theoremuneq12i 3072 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
A = B    &   𝐶 = 𝐷       (A𝐶) = (B𝐷)
 
Theoremuneq1d 3073 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
(φA = B)       (φ → (A𝐶) = (B𝐶))
 
Theoremuneq2d 3074 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
(φA = B)       (φ → (𝐶A) = (𝐶B))
 
Theoremuneq12d 3075 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))
 
Theoremnfun 3076 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
xA    &   xB       x(AB)
 
Theoremunass 3077 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((AB) ∪ 𝐶) = (A ∪ (B𝐶))
 
Theoremun12 3078 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
(A ∪ (B𝐶)) = (B ∪ (A𝐶))
 
Theoremun23 3079 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((AB) ∪ 𝐶) = ((A𝐶) ∪ B)
 
Theoremun4 3080 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
((AB) ∪ (𝐶𝐷)) = ((A𝐶) ∪ (B𝐷))
 
Theoremunundi 3081 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
(A ∪ (B𝐶)) = ((AB) ∪ (A𝐶))
 
Theoremunundir 3082 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
((AB) ∪ 𝐶) = ((A𝐶) ∪ (B𝐶))
 
Theoremssun1 3083 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
A ⊆ (AB)
 
Theoremssun2 3084 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
A ⊆ (BA)
 
Theoremssun3 3085 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
(ABA ⊆ (B𝐶))
 
Theoremssun4 3086 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
(ABA ⊆ (𝐶B))
 
Theoremelun1 3087 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
(A BA (B𝐶))
 
Theoremelun2 3088 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
(A BA (𝐶B))
 
Theoremunss1 3089 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(AB → (A𝐶) ⊆ (B𝐶))
 
Theoremssequn1 3090 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(AB ↔ (AB) = B)
 
Theoremunss2 3091 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
(AB → (𝐶A) ⊆ (𝐶B))
 
Theoremunss12 3092 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
((AB 𝐶𝐷) → (A𝐶) ⊆ (B𝐷))
 
Theoremssequn2 3093 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
(AB ↔ (BA) = B)
 
Theoremunss 3094 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
((A𝐶 B𝐶) ↔ (AB) ⊆ 𝐶)
 
Theoremunssi 3095 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
A𝐶    &   B𝐶       (AB) ⊆ 𝐶
 
Theoremunssd 3096 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φA𝐶)    &   (φB𝐶)       (φ → (AB) ⊆ 𝐶)
 
Theoremunssad 3097 If (AB) is contained in 𝐶, so is A. One-way deduction form of unss 3094. Partial converse of unssd 3096. (Contributed by David Moews, 1-May-2017.)
(φ → (AB) ⊆ 𝐶)       (φA𝐶)
 
Theoremunssbd 3098 If (AB) is contained in 𝐶, so is B. One-way deduction form of unss 3094. Partial converse of unssd 3096. (Contributed by David Moews, 1-May-2017.)
(φ → (AB) ⊆ 𝐶)       (φB𝐶)
 
Theoremssun 3099 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
((AB A𝐶) → A ⊆ (B𝐶))
 
Theoremrexun 3100 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
(x (AB)φ ↔ (x A φ x B φ))
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