Home Metamath Proof ExplorerTheorem List (p. 41 of 458) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28800) Hilbert Space Explorer (28801-30323) Users' Mathboxes (30324-45724)

Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempssss 4001 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
(𝐴𝐵𝐴𝐵)

Theorempssne 4002 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
(𝐴𝐵𝐴𝐵)

Theorempssssd 4003 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
(𝜑𝐴𝐵)       (𝜑𝐴𝐵)

Theorempssned 4004 Proper subclasses are unequal. Deduction form of pssne 4002. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑𝐴𝐵)

Theoremsspss 4005 Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
(𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Theorempssirr 4006 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
¬ 𝐴𝐴

Theorempssn2lp 4007 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.)
¬ (𝐴𝐵𝐵𝐴)

Theoremsspsstri 4008 Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Theoremssnpss 4009 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → ¬ 𝐵𝐴)

Theorempsstr 4010 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremsspsstr 4011 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theorempsssstr 4012 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theorempsstrd 4013 Proper subclass inclusion is transitive. Deduction form of psstr 4010. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsspsstrd 4014 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4011. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theorempsssstrd 4015 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4012. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremnpss 4016 A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3907. (Contributed by Mario Carneiro, 15-May-2015.)
𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Theoremssnelpss 4017 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
(𝐴𝐵 → ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → 𝐴𝐵))

Theoremssnelpssd 4018 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4017. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)    &   (𝜑 → ¬ 𝐶𝐴)       (𝜑𝐴𝐵)

Theoremssexnelpss 4019* If there is an element of a class which is not contained in a subclass, the subclass is a proper subclass. (Contributed by AV, 29-Jan-2020.)
((𝐴𝐵 ∧ ∃𝑥𝐵 𝑥𝐴) → 𝐴𝐵)

2.1.13  The difference, union, and intersection of two classes

2.1.13.1  The difference of two classes

Theoremdfdif3 4020* Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.)
(𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}

Theoremdifeq1 4021 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremdifeq2 4022 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremdifeq12 4023 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremdifeq1i 4024 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremdifeq2i 4025 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremdifeq12i 4026 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremdifeq1d 4027 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremdifeq2d 4028 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremdifeq12d 4029 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremdifeqri 4030* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶

Theoremnfdif 4031 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 4032 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)

Theoremeldifn 4033 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
(𝐴 ∈ (𝐵𝐶) → ¬ 𝐴𝐶)

Theoremelndif 4034 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶𝐵))

Theoremneldif 4035 Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵𝐶)) → 𝐴𝐶)

Theoremdifdif 4036 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
(𝐴 ∖ (𝐵𝐴)) = 𝐴

Theoremdifss 4037 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵) ⊆ 𝐴

Theoremdifssd 4038 A difference of two classes is contained in the minuend. Deduction form of difss 4037. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐴)

Theoremdifss2 4039 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 4040 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4039. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ⊆ (𝐵𝐶))       (𝜑𝐴𝐵)

Theoremssdifss 4041 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
(𝐴𝐵 → (𝐴𝐶) ⊆ 𝐵)

Theoremddif 4042 Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
(V ∖ (V ∖ 𝐴)) = 𝐴

Theoremssconb 4043 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))

Theoremsscon 4044 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → (𝐶𝐵) ⊆ (𝐶𝐴))

Theoremssdif 4045 Difference law for subsets. (Contributed by NM, 28-May-1998.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssdifd 4046 If 𝐴 is contained in 𝐵, then (𝐴𝐶) is contained in (𝐵𝐶). Deduction form of ssdif 4045. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremsscond 4047 If 𝐴 is contained in 𝐵, then (𝐶𝐵) is contained in (𝐶𝐴). Deduction form of sscon 4044. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐵) ⊆ (𝐶𝐴))

Theoremssdifssd 4048 If 𝐴 is contained in 𝐵, then (𝐴𝐶) is also contained in 𝐵. Deduction form of ssdifss 4041. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ 𝐵)

Theoremssdif2d 4049 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 4050 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
(∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)

Theoremrexdifi 4051 Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.)
((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴𝐵)𝜑)

Theoremcomplss 4052 Complementation reverses inclusion. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 19-Mar-2021.)
(𝐴𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))

Theoremcompleq 4053 Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.)
(𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵))

2.1.13.2  The union of two classes

Theoremelun 4054 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Theoremelunnel1 4055 A member of a union that is not member of the first class, is member of the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐵) → 𝐴𝐶)

Theoremuneqri 4056* Inference from membership to union. (Contributed by NM, 21-Jun-1993.)
((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶

Theoremunidm 4057 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 21-Jun-1993.)
(𝐴𝐴) = 𝐴

Theoremuncom 4058 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 4059 If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4059 was automatically derived from equncomVD 41947 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
(𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))

Theoremequncomi 4060 Inference form of equncom 4059. equncomi 4060 was automatically derived from equncomiVD 41948 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
𝐴 = (𝐵𝐶)       𝐴 = (𝐶𝐵)

Theoremuneq1 4061 Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremuneq2 4062 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremuneq12 4063 Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremuneq1i 4064 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremuneq2i 4065 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremuneq12i 4066 Equality inference for the 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 4067 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremuneq2d 4068 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremuneq12d 4069 Equality deduction for the 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 4070 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 4071 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 4072 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))

Theoremun23 4073 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)

Theoremun4 4074 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))

Theoremunundi 4075 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremunundir 4076 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremssun1 4077 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
𝐴 ⊆ (𝐴𝐵)

Theoremssun2 4078 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
𝐴 ⊆ (𝐵𝐴)

Theoremssun3 4079 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ⊆ (𝐵𝐶))

Theoremssun4 4080 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
(𝐴𝐵𝐴 ⊆ (𝐶𝐵))

Theoremelun1 4081 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐵𝐶))

Theoremelun2 4082 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐶𝐵))

Theoremelunant 4083 A statement is true for every element of the union of a pair of classes if and only if it is true for every element of the first class and for every element of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
((𝐶 ∈ (𝐴𝐵) → 𝜑) ↔ ((𝐶𝐴𝜑) ∧ (𝐶𝐵𝜑)))

Theoremunss1 4084 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssequn1 4085 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 4086 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremunss12 4087 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremssequn2 4088 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)

Theoremunss 4089 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)

Theoremunssi 4090 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
𝐴𝐶    &   𝐵𝐶       (𝐴𝐵) ⊆ 𝐶

Theoremunssd 4091 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 4092 If (𝐴𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4089. Partial converse of unssd 4091. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐴𝐶)

Theoremunssbd 4093 If (𝐴𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 4089. Partial converse of unssd 4091. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐵𝐶)

Theoremssun 4094 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))

Theoremrexun 4095 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
(∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑))

Theoremralunb 4096 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑))

Theoremralun 4097 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑) → ∀𝑥 ∈ (𝐴𝐵)𝜑)

2.1.13.3  The intersection of two classes

Theoremelini 4098 Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∈ (𝐵𝐶)

Theoremelind 4099 Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝑋𝐴)    &   (𝜑𝑋𝐵)       (𝜑𝑋 ∈ (𝐴𝐵))

Theoremelinel1 4100 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45724
 Copyright terms: Public domain < Previous  Next >