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Theorem List for Metamath Proof Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcsb2 3501* Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that 𝑥 can be free in 𝐵 but cannot occur in 𝐴. (Contributed by NM, 2-Dec-2013.)
𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴𝑦𝐵)}

Theoremcsbeq1 3502 Analogue of dfsbcq 3404 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝐴 = 𝐵𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)

Theoremcsbeq2 3503 Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
(∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)

Theoremcbvcsb 3504 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑦𝐶    &   𝑥𝐷    &   (𝑥 = 𝑦𝐶 = 𝐷)       𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷

Theoremcbvcsbv 3505* Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝑥 = 𝑦𝐵 = 𝐶)       𝐴 / 𝑥𝐵 = 𝐴 / 𝑦𝐶

Theoremcsbeq1d 3506 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
(𝜑𝐴 = 𝐵)       (𝜑𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)

Theoremcsbid 3507 Analogue of sbid 2100 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
𝑥 / 𝑥𝐴 = 𝐴

Theoremcsbeq1a 3508 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝑥 = 𝐴𝐵 = 𝐴 / 𝑥𝐵)

Theoremcsbco 3509* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵

Theoremcsbtt 3510 Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
((𝐴𝑉𝑥𝐵) → 𝐴 / 𝑥𝐵 = 𝐵)

Theoremcsbconstgf 3511 Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by NM, 10-Nov-2005.)
𝑥𝐵       (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)

Theoremcsbconstg 3512* Substitution doesn't affect a constant 𝐵 (in which 𝑥 does not occur). csbconstgf 3511 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
(𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)

Theoremnfcsb1d 3513 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑𝑥𝐴 / 𝑥𝐵)

Theoremnfcsb1 3514 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴       𝑥𝐴 / 𝑥𝐵

Theoremnfcsb1v 3515* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴 / 𝑥𝐵

Theoremnfcsbd 3516 Deduction version of nfcsb 3517. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴 / 𝑦𝐵)

Theoremnfcsb 3517 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴 / 𝑦𝐵

Theoremcsbhypf 3518* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3226 for class substitution version. (Contributed by NM, 19-Dec-2008.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)

Theoremcsbiebt 3519* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3523.) (Contributed by NM, 11-Nov-2005.)
((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))

Theoremcsbiedf 3520* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
𝑥𝜑    &   (𝜑𝑥𝐶)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑𝐴 / 𝑥𝐵 = 𝐶)

Theoremcsbieb 3521* Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
𝐴 ∈ V    &   𝑥𝐶       (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶)

Theoremcsbiebg 3522* Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝐶       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))

Theoremcsbiegf 3523* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝐴𝑉𝑥𝐶)    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)

Theoremcsbief 3524* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝐴 ∈ V    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       𝐴 / 𝑥𝐵 = 𝐶

Theoremcsbie 3525* Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)       𝐴 / 𝑥𝐵 = 𝐶

Theoremcsbied 3526* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑𝐴 / 𝑥𝐵 = 𝐶)

Theoremcsbied2 3527* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐴𝑉)    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑥 = 𝐵) → 𝐶 = 𝐷)       (𝜑𝐴 / 𝑥𝐶 = 𝐷)

Theoremcsbie2t 3528* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3529). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)

Theoremcsbie2 3529* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)       𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷

Theoremcsbie2g 3530* Conversion of implicit substitution to explicit class substitution. This version of csbie 3525 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   (𝑦 = 𝐴𝐶 = 𝐷)       (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)

Theoremcbvralcsf 3531 A more general version of cbvralf 3141 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))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐵 𝜓)

Theoremcbvrexcsf 3532 A more general version of cbvrexf 3142 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))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)

Theoremcbvreucsf 3533 A more general version of cbvreuv 3149 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
𝑦𝐴    &   𝑥𝐵    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐵 𝜓)

Theoremcbvrabcsf 3534 A more general version of cbvrab 3171 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
𝑦𝐴    &   𝑥𝐵    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐵𝜓}

Theoremcbvralv2 3535* 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)       (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒)

Theoremcbvrexv2 3536* 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)       (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒)

2.1.11  Define basic set operations and relations

Syntaxcdif 3537 Extend class notation to include class difference (read: "𝐴 minus 𝐵").
class (𝐴𝐵)

Syntaxcun 3538 Extend class notation to include union of two classes (read: "𝐴 union 𝐵").
class (𝐴𝐵)

Syntaxcin 3539 Extend class notation to include the intersection of two classes (read: "𝐴 intersect 𝐵").
class (𝐴𝐵)

Syntaxwss 3540 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 3541 Extend wff notation with proper subclass relation.
wff 𝐴𝐵

Theoremdifjust 3542* Soundness justification theorem for df-dif 3543. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴 ∧ ¬ 𝑦𝐵)}

Definitiondf-dif 3543* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ({1, 3} ∖ {1, 8}) = {3} (ex-dif 26672). Contrast this operation with union (𝐴𝐵) (df-un 3545) and intersection (𝐴𝐵) (df-in 3547). 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 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}

Theoremunjust 3544* Soundness justification theorem for df-un 3545. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}

Definitiondf-un 3545* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∪ {1, 8}) = {1, 3, 8} (ex-un 26673). Contrast this operation with difference (𝐴𝐵) (df-dif 3543) and intersection (𝐴𝐵) (df-in 3547). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 3821. For union defined in terms of intersection, see dfun3 3824. (Contributed by NM, 23-Aug-1993.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}

Theoreminjust 3546* Soundness justification theorem for df-in 3547. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}

Definitiondf-in 3547* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 26674). Contrast this operation with union (𝐴𝐵) (df-un 3545) and difference (𝐴𝐵) (df-dif 3543). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3822 and dfin4 3826. For intersection defined in terms of union, see dfin3 3825. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}

Theoremdfin5 3548* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
(𝐴𝐵) = {𝑥𝐴𝑥𝐵}

Theoremdfdif2 3549* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
(𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}

Theoremeldif 3550 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))

Theoremeldifd 3551 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3550. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐴𝐶)       (𝜑𝐴 ∈ (𝐵𝐶))

Theoremeldifad 3552 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3550. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑𝐴𝐵)

Theoremeldifbd 3553 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3550. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑 → ¬ 𝐴𝐶)

2.1.12  Subclasses and subsets

Definitiondf-ss 3554 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example, {1, 2} ⊆ {1, 2, 3} (ex-ss 26676). Note that 𝐴𝐴 (proved in ssid 3587). Contrast this relationship with the relationship 𝐴𝐵 (as will be defined in df-pss 3556). For a more traditional definition, but requiring a dummy variable, see dfss2 3557. Other possible definitions are given by dfss3 3558, dfss4 3820, sspss 3668, ssequn1 3745, ssequn2 3748, sseqin2 3779, and ssdif0 3896. (Contributed by NM, 27-Apr-1994.)
(𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)

Theoremdfss 3555 Variant of subclass definition df-ss 3554. (Contributed by NM, 21-Jun-1993.)
(𝐴𝐵𝐴 = (𝐴𝐵))

Definitiondf-pss 3556 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. For example, {1, 2} ⊊ {1, 2, 3} (ex-pss 26677). Note that ¬ 𝐴𝐴 (proved in pssirr 3669). Contrast this relationship with the relationship 𝐴𝐵 (as defined in df-ss 3554). Other possible definitions are given by dfpss2 3654 and dfpss3 3655. (Contributed by NM, 7-Feb-1996.)
(𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))

Theoremdfss2 3557* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
(𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremdfss3 3558* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)

Theoremdfss2f 3559 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       (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremdfss3f 3560 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)

Theoremnfss 3561 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 3562 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))

Theoremssel2 3563 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
((𝐴𝐵𝐶𝐴) → 𝐶𝐵)

Theoremsseli 3564 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)

Theoremsselii 3565 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
𝐴𝐵    &   𝐶𝐴       𝐶𝐵

Theoremsseldi 3566 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
𝐴𝐵    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)

Theoremsseld 3567 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremsselda 3568 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
(𝜑𝐴𝐵)       ((𝜑𝐶𝐴) → 𝐶𝐵)

Theoremsseldd 3569 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)

Theoremssneld 3570 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 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))

Theoremssneldd 3571 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 → ¬ 𝐶𝐵)       (𝜑 → ¬ 𝐶𝐴)

Theoremssriv 3572* Inference rule based on subclass definition. (Contributed by NM, 21-Jun-1993.)
(𝑥𝐴𝑥𝐵)       𝐴𝐵

Theoremssrd 3573 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)

Theoremssrdv 3574* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
(𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)

Theoremsstr2 3575 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝐴𝐵 → (𝐵𝐶𝐴𝐶))

Theoremsstr 3576 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremsstri 3577 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
𝐴𝐵    &   𝐵𝐶       𝐴𝐶

Theoremsstrd 3578 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsyl5ss 3579 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
𝐴𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsyl6ss 3580 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)

Theoremsylan9ss 3581 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜑𝜓) → 𝐴𝐶)

Theoremsylan9ssr 3582 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜓𝜑) → 𝐴𝐶)

Theoremeqss 3583 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 21-May-1993.)
(𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))

Theoremeqssi 3584 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
𝐴𝐵    &   𝐵𝐴       𝐴 = 𝐵

Theoremeqssd 3585 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = 𝐵)

Theoremeqrd 3586 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)

Theoremssid 3587 Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
𝐴𝐴

Theoremssv 3588 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
𝐴 ⊆ V

Theoremsseq1 3589 Equality theorem for subclasses. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremsseq2 3590 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremsseq12 3591 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))

Theoremsseq1i 3592 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)

Theoremsseq2i 3593 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)

Theoremsseq12i 3594 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 3595 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))

Theoremsseq2d 3596 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremsseq12d 3597 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theoremeqsstri 3598 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶

Theoremeqsstr3i 3599 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
𝐵 = 𝐴    &   𝐵𝐶       𝐴𝐶

Theoremsseqtri 3600 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶

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