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Theorem List for Metamath Proof Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrmob2 3401* Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.)
(𝑥 = 𝐵 → (𝜓𝜒))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∃*𝑥𝐴 𝜓)    &   (𝜑𝑥𝐴)    &   (𝜑𝜓)       (𝜑 → (𝑥 = 𝐵𝜒))
 
Theoremrmoi2 3402* Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.)
(𝑥 = 𝐵 → (𝜓𝜒))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∃*𝑥𝐴 𝜓)    &   (𝜑𝑥𝐴)    &   (𝜑𝜓)    &   (𝜑𝜒)       (𝜑𝑥 = 𝐵)
 
2.1.10  Proper substitution of classes for sets into classes
 
Syntaxcsb 3403 Extend class notation to include the proper substitution of a class for a set into another class.
class 𝐴 / 𝑥𝐵
 
Definitiondf-csb 3404* Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 3306, to prevent ambiguity. Theorem sbcel1g 3842 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsb 3859 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
 
Theoremcsb2 3405* 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 3406 Analogue of dfsbcq 3308 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝐴 = 𝐵𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)
 
Theoremcsbeq2 3407 Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
(∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
 
Theoremcbvcsb 3408 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 3409* 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 3410 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
(𝜑𝐴 = 𝐵)       (𝜑𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐶)
 
Theoremcsbid 3411 Analogue of sbid 2133 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
𝑥 / 𝑥𝐴 = 𝐴
 
Theoremcsbeq1a 3412 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(𝑥 = 𝐴𝐵 = 𝐴 / 𝑥𝐵)
 
Theoremcsbco 3413* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
 
Theoremcsbtt 3414 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 3415 Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by NM, 10-Nov-2005.)
𝑥𝐵       (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
 
Theoremcsbconstg 3416* Substitution doesn't affect a constant 𝐵 (in which 𝑥 does not occur). csbconstgf 3415 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
(𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
 
Theoremnfcsb1d 3417 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑𝑥𝐴 / 𝑥𝐵)
 
Theoremnfcsb1 3418 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴       𝑥𝐴 / 𝑥𝐵
 
Theoremnfcsb1v 3419* 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 3420 Deduction version of nfcsb 3421. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴 / 𝑦𝐵)
 
Theoremnfcsb 3421 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴 / 𝑦𝐵
 
Theoremcsbhypf 3422* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3130 for class substitution version. (Contributed by NM, 19-Dec-2008.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
 
Theoremcsbiebt 3423* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3427.) (Contributed by NM, 11-Nov-2005.)
((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
 
Theoremcsbiedf 3424* 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 3425* 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 3426* 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 3427* 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 3428* 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 3429* 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 3430* 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 3431* 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 3432* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3433). (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 3433* 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 3434* Conversion of implicit substitution to explicit class substitution. This version of csbie 3429 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 3435 A more general version of cbvralf 3045 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 3436 A more general version of cbvrexf 3046 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 3437 A more general version of cbvreuv 3053 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 3438 A more general version of cbvrab 3075 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 3439* 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 3440* 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 3441 Extend class notation to include class difference (read: "𝐴 minus 𝐵").
class (𝐴𝐵)
 
Syntaxcun 3442 Extend class notation to include union of two classes (read: "𝐴 union 𝐵").
class (𝐴𝐵)
 
Syntaxcin 3443 Extend class notation to include the intersection of two classes (read: "𝐴 intersect 𝐵").
class (𝐴𝐵)
 
Syntaxwss 3444 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 3445 Extend wff notation with proper subclass relation.
wff 𝐴𝐵
 
Theoremdifjust 3446* Soundness justification theorem for df-dif 3447. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴 ∧ ¬ 𝑦𝐵)}
 
Definitiondf-dif 3447* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ({1, 3} ∖ {1, 8}) = {3} (ex-dif 26410). Contrast this operation with union (𝐴𝐵) (df-un 3449) and intersection (𝐴𝐵) (df-in 3451). 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 3448* Soundness justification theorem for df-un 3449. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
 
Definitiondf-un 3449* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∪ {1, 8}) = {1, 3, 8} (ex-un 26411). Contrast this operation with difference (𝐴𝐵) (df-dif 3447) and intersection (𝐴𝐵) (df-in 3451). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 3724. For union defined in terms of intersection, see dfun3 3727. (Contributed by NM, 23-Aug-1993.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
 
Theoreminjust 3450* Soundness justification theorem for df-in 3451. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
 
Definitiondf-in 3451* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 26412). Contrast this operation with union (𝐴𝐵) (df-un 3449) and difference (𝐴𝐵) (df-dif 3447). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3725 and dfin4 3729. For intersection defined in terms of union, see dfin3 3728. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
 
Theoremdfin5 3452* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
(𝐴𝐵) = {𝑥𝐴𝑥𝐵}
 
Theoremdfdif2 3453* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
(𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
 
Theoremeldif 3454 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
 
Theoremeldifd 3455 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3454. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐴𝐶)       (𝜑𝐴 ∈ (𝐵𝐶))
 
Theoremeldifad 3456 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3454. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑𝐴𝐵)
 
Theoremeldifbd 3457 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3454. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑 → ¬ 𝐴𝐶)
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 3458 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example, {1, 2} ⊆ {1, 2, 3} (ex-ss 26414). Note that 𝐴𝐴 (proved in ssid 3491). Contrast this relationship with the relationship 𝐴𝐵 (as will be defined in df-pss 3460). For a more traditional definition, but requiring a dummy variable, see dfss2 3461. Other possible definitions are given by dfss3 3462, dfss4 3723, sspss 3572, ssequn1 3649, ssequn2 3652, sseqin2 3682, and ssdif0 3799. (Contributed by NM, 27-Apr-1994.)
(𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
 
Theoremdfss 3459 Variant of subclass definition df-ss 3458. (Contributed by NM, 21-Jun-1993.)
(𝐴𝐵𝐴 = (𝐴𝐵))
 
Definitiondf-pss 3460 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. For example, {1, 2} ⊊ {1, 2, 3} (ex-pss 26415). Note that ¬ 𝐴𝐴 (proved in pssirr 3573). Contrast this relationship with the relationship 𝐴𝐵 (as defined in df-ss 3458). Other possible definitions are given by dfpss2 3558 and dfpss3 3559. (Contributed by NM, 7-Feb-1996.)
(𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
 
Theoremdfss2 3461* 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 3462* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremdfss2f 3463 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 3464 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 3465 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 3466 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremssel2 3467 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
((𝐴𝐵𝐶𝐴) → 𝐶𝐵)
 
Theoremsseli 3468 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremsselii 3469 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
𝐴𝐵    &   𝐶𝐴       𝐶𝐵
 
Theoremsseldi 3470 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
𝐴𝐵    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)
 
Theoremsseld 3471 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremsselda 3472 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
(𝜑𝐴𝐵)       ((𝜑𝐶𝐴) → 𝐶𝐵)
 
Theoremsseldd 3473 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)
 
Theoremssneld 3474 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 3475 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 3476* Inference rule based on subclass definition. (Contributed by NM, 21-Jun-1993.)
(𝑥𝐴𝑥𝐵)       𝐴𝐵
 
Theoremssrd 3477 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
 
Theoremssrdv 3478* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
(𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
 
Theoremsstr2 3479 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 3480 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsstri 3481 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
𝐴𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremsstrd 3482 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl5ss 3483 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
𝐴𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl6ss 3484 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsylan9ss 3485 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜑𝜓) → 𝐴𝐶)
 
Theoremsylan9ssr 3486 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜓𝜑) → 𝐴𝐶)
 
Theoremeqss 3487 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 21-May-1993.)
(𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
 
Theoremeqssi 3488 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
𝐴𝐵    &   𝐵𝐴       𝐴 = 𝐵
 
Theoremeqssd 3489 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = 𝐵)
 
Theoremeqrd 3490 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremssid 3491 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 3492 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
𝐴 ⊆ V
 
Theoremsseq1 3493 Equality theorem for subclasses. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremsseq2 3494 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremsseq12 3495 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
 
Theoremsseq1i 3496 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremsseq2i 3497 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremsseq12i 3498 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 3499 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremsseq2d 3500 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
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