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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcsbconstgi 3901* The proper substitution of a class for a variable in another variable does not modify it, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝐴 ∈ V       𝐴 / 𝑥𝑦 = 𝑦
 
Theoremnfcsb1d 3902 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑𝑥𝐴 / 𝑥𝐵)
 
Theoremnfcsb1 3903 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴       𝑥𝐴 / 𝑥𝐵
 
Theoremnfcsb1v 3904* 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 3905 Deduction version of nfcsb 3907. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴 / 𝑦𝐵)
 
Theoremnfcsbw 3906* Version of nfcsb 3907 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴 / 𝑦𝐵
 
Theoremnfcsb 3907 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴 / 𝑦𝐵
 
Theoremcsbhypf 3908* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3550 for class substitution version. (Contributed by NM, 19-Dec-2008.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐶)
 
Theoremcsbiebt 3909* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3913.) (Contributed by NM, 11-Nov-2005.)
((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
 
Theoremcsbiedf 3910* 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 3911* 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 3912* 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 3913* 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 3914* 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 3915* 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 3916* 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 3917* 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 3918* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3919). (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 3919* 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 3920* Conversion of implicit substitution to explicit class substitution. This version of csbie 3915 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)
 
Theoremcbvrabcsfw 3921* Version of cbvrabcsf 3925 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑦𝐴    &   𝑥𝐵    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
 
Theoremcbvralcsf 3922 A more general version of cbvralf 3437 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 3923 A more general version of cbvrexf 3438 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 3924 A more general version of cbvreuv 3452 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 3925 A more general version of cbvrab 3488 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 3926* 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 3927* 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)
 
Theoremvtocl2dOLD 3928* Obsolete version of vtocl2d 3555 as of 19-Oct-2023. (Contributed by Thierry Arnoux, 25-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))    &   (𝜑𝜓)       (𝜑𝜒)
 
Theoremrspc2vd 3929* Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class 𝐷 for the second set variable 𝑦 may depend on the first set variable 𝑥. (Contributed by AV, 29-Mar-2021.)
(𝑥 = 𝐴 → (𝜃𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝐶)    &   ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐸)    &   (𝜑𝐵𝐸)       (𝜑 → (∀𝑥𝐶𝑦𝐷 𝜃𝜓))
 
2.1.11  Define basic set operations and relations
 
Syntaxcdif 3930 Extend class notation to include class difference (read: "𝐴 minus 𝐵").
class (𝐴𝐵)
 
Syntaxcun 3931 Extend class notation to include union of two classes (read: "𝐴 union 𝐵").
class (𝐴𝐵)
 
Syntaxcin 3932 Extend class notation to include the intersection of two classes (read: "𝐴 intersect 𝐵").
class (𝐴𝐵)
 
Syntaxwss 3933 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 3934 Extend wff notation with proper subclass relation.
wff 𝐴𝐵
 
Theoremdifjust 3935* Soundness justification theorem for df-dif 3936. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴 ∧ ¬ 𝑦𝐵)}
 
Definitiondf-dif 3936* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ({1, 3} ∖ {1, 8}) = {3} (ex-dif 28129). Contrast this operation with union (𝐴𝐵) (df-un 3938) and intersection (𝐴𝐵) (df-in 3940). 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 3937* Soundness justification theorem for df-un 3938. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
 
Definitiondf-un 3938* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∪ {1, 8}) = {1, 3, 8} (ex-un 28130). Contrast this operation with difference (𝐴𝐵) (df-dif 3936) and intersection (𝐴𝐵) (df-in 3940). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 4233. For union defined in terms of intersection, see dfun3 4239. (Contributed by NM, 23-Aug-1993.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
 
Theoreminjust 3939* Soundness justification theorem for df-in 3940. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
 
Definitiondf-in 3940* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 28131). Contrast this operation with union (𝐴𝐵) (df-un 3938) and difference (𝐴𝐵) (df-dif 3936). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4234 and dfin4 4241. For intersection defined in terms of union, see dfin3 4240. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
 
Theoremdfin5 3941* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
(𝐴𝐵) = {𝑥𝐴𝑥𝐵}
 
Theoremdfdif2 3942* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
(𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
 
Theoremeldif 3943 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
 
Theoremeldifd 3944 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3943. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐴𝐶)       (𝜑𝐴 ∈ (𝐵𝐶))
 
Theoremeldifad 3945 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3943. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑𝐴𝐵)
 
Theoremeldifbd 3946 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3943. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑 → ¬ 𝐴𝐶)
 
Theoremelneeldif 3947 The elements of a set difference and the minuend are not equal. (Contributed by AV, 21-Oct-2023.)
((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑋𝑌)
 
Theoremvelcomp 3948 Characterization of setvar elements of the complement of a class. (Contributed by Andrew Salmon, 15-Jul-2011.)
(𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 3949 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example, {1, 2} ⊆ {1, 2, 3} (ex-ss 28133). Note that 𝐴𝐴 (proved in ssid 3986). Contrast this relationship with the relationship 𝐴𝐵 (as will be defined in df-pss 3951). For a more traditional definition, but requiring a dummy variable, see dfss2 3952. Other possible definitions are given by dfss3 3953, dfss4 4232, sspss 4073, ssequn1 4153, ssequn2 4156, sseqin2 4189, and ssdif0 4320.

We prefer the label "ss" ("subset") for , despite the fact that it applies to classes. It is much more common to refer to this as the subset relation than subclass, especially since most of the time the arguments are in fact sets (and for pragmatic reasons we don't want to need to use different operations for sets). The way set.mm is set up, many things are technically classes despite morally (and provably) being sets, like 1 (cf. df-1 10533 and 1ex 10625) or ( cf. df-r 10535 and reex 10616). This has to do with the fact that there are no "set expressions": classes are expressions but there are only set variables in set.mm (cf. https://us.metamath.org/downloads/grammar-ambiguity.txt 10616). This is why we use both for subclass relations and for subset relations and call it "subset". (Contributed by NM, 27-Apr-1994.)

(𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
 
Theoremdfss 3950 Variant of subclass definition df-ss 3949. (Contributed by NM, 21-Jun-1993.)
(𝐴𝐵𝐴 = (𝐴𝐵))
 
Definitiondf-pss 3951 Define proper subclass (or strict subclass) relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. For example, {1, 2} ⊊ {1, 2, 3} (ex-pss 28134). Note that ¬ 𝐴𝐴 (proved in pssirr 4074). Contrast this relationship with the relationship 𝐴𝐵 (as defined in df-ss 3949). Other possible definitions are given by dfpss2 4059 and dfpss3 4060. (Contributed by NM, 7-Feb-1996.)
(𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
 
Theoremdfss2 3952* 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 3953* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremdfss6 3954* Alternate definition of subclass relationship. (Contributed by RP, 16-Apr-2020.)
(𝐴𝐵 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
 
Theoremdfss2f 3955 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.) Avoid ax-13 2381. (Revised by Gino Giotto, 19-May-2023.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremdfss3f 3956 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 3957 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 3958 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremssel2 3959 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
((𝐴𝐵𝐶𝐴) → 𝐶𝐵)
 
Theoremsseli 3960 Membership implication from subclass relationship. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremsselii 3961 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
𝐴𝐵    &   𝐶𝐴       𝐶𝐵
 
Theoremsseldi 3962 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
𝐴𝐵    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)
 
Theoremsseld 3963 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremsselda 3964 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
(𝜑𝐴𝐵)       ((𝜑𝐶𝐴) → 𝐶𝐵)
 
Theoremsseldd 3965 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)
 
Theoremssneld 3966 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 3967 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 3968* Inference based on subclass definition. (Contributed by NM, 21-Jun-1993.)
(𝑥𝐴𝑥𝐵)       𝐴𝐵
 
Theoremssrd 3969 Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
 
Theoremssrdv 3970* Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.)
(𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
 
Theoremsstr2 3971 Transitivity of subclass relationship. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝐴𝐵 → (𝐵𝐶𝐴𝐶))
 
Theoremsstr 3972 Transitivity of subclass relationship. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsstri 3973 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
𝐴𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremsstrd 3974 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsstrid 3975 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
𝐴𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsstrdi 3976 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsylan9ss 3977 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜑𝜓) → 𝐴𝐶)
 
Theoremsylan9ssr 3978 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜓𝜑) → 𝐴𝐶)
 
Theoremeqss 3979 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 21-May-1993.)
(𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
 
Theoremeqssi 3980 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
𝐴𝐵    &   𝐵𝐴       𝐴 = 𝐵
 
Theoremeqssd 3981 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = 𝐵)
 
Theoremsssseq 3982 If a class is a subclass of another class, then the classes are equal if and only if the other class is a subclass of the first class. (Contributed by AV, 23-Dec-2020.)
(𝐵𝐴 → (𝐴𝐵𝐴 = 𝐵))
 
Theoremeqrd 3983 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.) (Proof shortened by BJ, 1-Dec-2021.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremeqri 3984 Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
𝑥𝐴    &   𝑥𝐵    &   (𝑥𝐴𝑥𝐵)       𝐴 = 𝐵
 
Theoremeqelssd 3985* Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥𝐵) → 𝑥𝐴)       (𝜑𝐴 = 𝐵)
 
Theoremssid 3986 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
 
Theoremssidd 3987 Weakening of ssid 3986. (Contributed by BJ, 1-Sep-2022.)
(𝜑𝐴𝐴)
 
Theoremssv 3988 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
𝐴 ⊆ V
 
Theoremsseq1 3989 Equality theorem for subclasses. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremsseq2 3990 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremsseq12 3991 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
 
Theoremsseq1i 3992 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremsseq2i 3993 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremsseq12i 3994 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 3995 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremsseq2d 3996 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremsseq12d 3997 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremeqsstri 3998 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqsstrri 3999 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
𝐵 = 𝐴    &   𝐵𝐶       𝐴𝐶
 
Theoremsseqtri 4000 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
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