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

Theoremrexss 3001* 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 3002 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
({xφ} ⊆ {xψ} ↔ x(φψ))

Theoremabss 3003* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({xφ} ⊆ Ax(φx A))

Theoremssab 3004* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
(A ⊆ {xφ} ↔ x(x Aφ))

Theoremssabral 3005* 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 3006 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
(φψ)       {xφ} ⊆ {xψ}

Theoremss2abdv 3007* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
(φ → (ψχ))       (φ → {xψ} ⊆ {xχ})

Theoremabssdv 3008* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(φ → (ψx A))       (φ → {xψ} ⊆ A)

Theoremabssi 3009* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(φx A)       {xφ} ⊆ A

Theoremss2rab 3010 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
({x Aφ} ⊆ {x Aψ} ↔ x A (φψ))

Theoremrabss 3011* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({x Aφ} ⊆ Bx A (φx B))

Theoremssrab 3012* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
(B ⊆ {x Aφ} ↔ (BA x B φ))

Theoremssrabdv 3013* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
(φBA)    &   ((φ x B) → ψ)       (φB ⊆ {x Aψ})

Theoremrabssdv 3014* 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 3015* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
((φ x A) → (ψχ))       (φ → {x Aψ} ⊆ {x Aχ})

Theoremss2rabi 3016 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
(x A → (φψ))       {x Aφ} ⊆ {x Aψ}

Theoremrabss2 3017* 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 3018* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
{x ∣ (x A φ)} ⊆ A

Theoremssrab2 3019* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
{x Aφ} ⊆ A

Theoremssrabeq 3020* 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 3021 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
{x Aφ} ⊆ {xφ}

Theoremuniiunlem 3022* 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 3023 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
(AB ↔ (AB ¬ A = B))

Theoremdfpss3 3024 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(AB ↔ (AB ¬ BA))

Theorempsseq1 3025 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(A = B → (A𝐶B𝐶))

Theorempsseq2 3026 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(A = B → (𝐶A𝐶B))

Theorempsseq1i 3027 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
A = B       (A𝐶B𝐶)

Theorempsseq2i 3028 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
A = B       (𝐶A𝐶B)

Theorempsseq12i 3029 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
A = B    &   𝐶 = 𝐷       (A𝐶B𝐷)

Theorempsseq1d 3030 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(φA = B)       (φ → (A𝐶B𝐶))

Theorempsseq2d 3031 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(φA = B)       (φ → (𝐶A𝐶B))

Theorempsseq12d 3032 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶B𝐷))

Theorempssss 3033 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
(ABAB)

Theorempssne 3034 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
(ABAB)

Theorempssssd 3035 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
(φAB)       (φAB)

Theorempssned 3036 Proper subclasses are unequal. Deduction form of pssne 3034. (Contributed by David Moews, 1-May-2017.)
(φAB)       (φAB)

Theoremsspssr 3037 Subclass in terms of proper subclass. (Contributed by Jim Kingdon, 16-Jul-2018.)
((AB A = B) → AB)

Theorempssirr 3038 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
¬ AA

Theorempssn2lp 3039 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 3040 Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)
((AB A = B BA) → (AB BA))

Theoremssnpss 3041 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(AB → ¬ BA)

Theoremsspssn 3042 Like pssn2lp 3039 but for subset and proper subset. (Contributed by Jim Kingdon, 17-Jul-2018.)
¬ (AB BA)

Theorempsstr 3043 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
((AB B𝐶) → A𝐶)

Theoremsspsstr 3044 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((AB B𝐶) → A𝐶)

Theorempsssstr 3045 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((AB B𝐶) → A𝐶)

Theorempsstrd 3046 Proper subclass inclusion is transitive. Deduction form of psstr 3043. (Contributed by David Moews, 1-May-2017.)
(φAB)    &   (φB𝐶)       (φA𝐶)

Theoremsspsstrd 3047 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3044. (Contributed by David Moews, 1-May-2017.)
(φAB)    &   (φB𝐶)       (φA𝐶)

Theorempsssstrd 3048 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3045. (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 3049 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(A = B → (A𝐶) = (B𝐶))

Theoremdifeq2 3050 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(A = B → (𝐶A) = (𝐶B))

Theoremdifeq12 3051 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
((A = B 𝐶 = 𝐷) → (A𝐶) = (B𝐷))

Theoremdifeq1i 3052 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
A = B       (A𝐶) = (B𝐶)

Theoremdifeq2i 3053 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
A = B       (𝐶A) = (𝐶B)

Theoremdifeq12i 3054 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
A = B    &   𝐶 = 𝐷       (A𝐶) = (B𝐷)

Theoremdifeq1d 3055 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
(φA = B)       (φ → (A𝐶) = (B𝐶))

Theoremdifeq2d 3056 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
(φA = B)       (φ → (𝐶A) = (𝐶B))

Theoremdifeq12d 3057 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))

Theoremdifeqri 3058* 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 3059 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 3060 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(A (B𝐶) → A B)

Theoremeldifn 3061 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
(A (B𝐶) → ¬ A 𝐶)

Theoremelndif 3062 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
(A B → ¬ A (𝐶B))

Theoremdifdif 3063 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
(A ∖ (BA)) = A

Theoremdifss 3064 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(AB) ⊆ A

Theoremdifssd 3065 A difference of two classes is contained in the minuend. Deduction form of difss 3064. (Contributed by David Moews, 1-May-2017.)
(φ → (AB) ⊆ A)

Theoremdifss2 3066 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 3067 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3066. (Contributed by David Moews, 1-May-2017.)
(φA ⊆ (B𝐶))       (φAB)

Theoremssdifss 3068 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
(AB → (A𝐶) ⊆ B)

Theoremddifnel 3069* 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 3169. (Contributed by Jim Kingdon, 21-Jul-2018.)
x (V ∖ A) → x A)       (V ∖ (V ∖ A)) = A

Theoremssconb 3070 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
((A𝐶 B𝐶) → (A ⊆ (𝐶B) ↔ B ⊆ (𝐶A)))

Theoremsscon 3071 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
(AB → (𝐶B) ⊆ (𝐶A))

Theoremssdif 3072 Difference law for subsets. (Contributed by NM, 28-May-1998.)
(AB → (A𝐶) ⊆ (B𝐶))

Theoremssdifd 3073 If A is contained in B, then (A𝐶) is contained in (B𝐶). Deduction form of ssdif 3072. (Contributed by David Moews, 1-May-2017.)
(φAB)       (φ → (A𝐶) ⊆ (B𝐶))

Theoremsscond 3074 If A is contained in B, then (𝐶B) is contained in (𝐶A). Deduction form of sscon 3071. (Contributed by David Moews, 1-May-2017.)
(φAB)       (φ → (𝐶B) ⊆ (𝐶A))

Theoremssdifssd 3075 If A is contained in B, then (A𝐶) is also contained in B. Deduction form of ssdifss 3068. (Contributed by David Moews, 1-May-2017.)
(φAB)       (φ → (A𝐶) ⊆ B)

Theoremssdif2d 3076 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 3077 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 3078 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 3079* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
((x A x B) ↔ x 𝐶)       (AB) = 𝐶

Theoremunidm 3080 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
(AA) = A

Theoremuncom 3081 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 3082 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 3083 Inference form of equncom 3082. (Contributed by Alan Sare, 18-Feb-2012.)
A = (B𝐶)       A = (𝐶B)

Theoremuneq1 3084 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
(A = B → (A𝐶) = (B𝐶))

Theoremuneq2 3085 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
(A = B → (𝐶A) = (𝐶B))

Theoremuneq12 3086 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
((A = B 𝐶 = 𝐷) → (A𝐶) = (B𝐷))

Theoremuneq1i 3087 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
A = B       (A𝐶) = (B𝐶)

Theoremuneq2i 3088 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
A = B       (𝐶A) = (𝐶B)

Theoremuneq12i 3089 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 3090 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
(φA = B)       (φ → (A𝐶) = (B𝐶))

Theoremuneq2d 3091 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
(φA = B)       (φ → (𝐶A) = (𝐶B))

Theoremuneq12d 3092 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 3093 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 3094 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 3095 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
(A ∪ (B𝐶)) = (B ∪ (A𝐶))

Theoremun23 3096 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((AB) ∪ 𝐶) = ((A𝐶) ∪ B)

Theoremun4 3097 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
((AB) ∪ (𝐶𝐷)) = ((A𝐶) ∪ (B𝐷))

Theoremunundi 3098 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
(A ∪ (B𝐶)) = ((AB) ∪ (A𝐶))

Theoremunundir 3099 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
((AB) ∪ 𝐶) = ((A𝐶) ∪ (B𝐶))

Theoremssun1 3100 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
A ⊆ (AB)

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