P. 40 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nfcsb 3901	Bound-variable hypothesis builder for substitution into a class. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker nfcsbw 3900 when possible. (Contributed by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
Theorem	csbhypf 3902*	Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3523 for class substitution version. (Contributed by NM, 19-Dec-2008.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbiebt 3903*	Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3907.) (Contributed by NM, 11-Nov-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
Theorem	csbiedf 3904*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbieb 3905*	Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbiebg 3906*	Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
Theorem	csbiegf 3907*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶)    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbief 3908*	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    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
Theorem	csbie 3909*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.) Reduce axiom usage. (Revised by GG, 15-Oct-2024.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
Theorem	csbied 3910*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) Reduce axiom usage. (Revised by GG, 15-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbied2 3911*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷)
Theorem	csbie2t 3912*	Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3913). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷)
Theorem	csbie2 3913*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷
Theorem	csbie2g 3914*	Conversion of implicit substitution to explicit class substitution. This version of csbie 3909 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    &   ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)
Theorem	cbvrabcsfw 3915*	Version of cbvrabcsf 3919 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by Andrew Salmon, 13-Jul-2011.) (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
Theorem	cbvralcsf 3916	A more general version of cbvralf 3339 that doesn't require 𝐴 and 𝐵 to be distinct from 𝑥 or 𝑦. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by Andrew Salmon, 13-Jul-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)
Theorem	cbvrexcsf 3917	A more general version of cbvrexf 3340 that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)
Theorem	cbvreucsf 3918	A more general version of cbvreuv 3410 that has no distinct variable restrictions. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by Andrew Salmon, 13-Jul-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓)
Theorem	cbvrabcsf 3919	A more general version of cbvrab 3458 with no distinct variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by Andrew Salmon, 13-Jul-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
Theorem	cbvralv2 3920*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)
Theorem	cbvrexv2 3921*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)
Theorem	rspc2vd 3922*	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 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐸)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐸)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → 𝜓))
2.1.11  Define basic set operations and relations
Syntax	cdif 3923	Extend class notation to include class difference (read: "𝐴 minus 𝐵").
class (𝐴 ∖ 𝐵)
Syntax	cun 3924	Extend class notation to include union of two classes (read: "𝐴 union 𝐵").
class (𝐴 ∪ 𝐵)
Syntax	cin 3925	Extend class notation to include the intersection of two classes (read: "𝐴 intersect 𝐵").
class (𝐴 ∩ 𝐵)
Syntax	wss 3926	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 𝐴 ⊆ 𝐵
Syntax	wpss 3927	Extend wff notation with proper subclass relation.
wff 𝐴 ⊊ 𝐵
Theorem	difjust 3928*	Soundness justification theorem for df-dif 3929. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)}
Definition	df-dif 3929*	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ({1, 3} ∖ {1, 8}) = {3} (ex-dif 30350). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3931) and intersection (𝐴 ∩ 𝐵) (df-in 3933). 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 𝐴 ∖ 𝐵. 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.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
Theorem	unjust 3930*	Soundness justification theorem for df-un 3931. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
Definition	df-un 3931*	Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∪ {1, 8}) = {1, 3, 8} (ex-un 30351). Contrast this operation with difference (𝐴 ∖ 𝐵) (df-dif 3929) and intersection (𝐴 ∩ 𝐵) (df-in 3933). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 4245. For union defined in terms of intersection, see dfun3 4251. (Contributed by NM, 23-Aug-1993.)
⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
Theorem	injust 3932*	Soundness justification theorem for df-in 3933. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
Definition	df-in 3933*	Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30352). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3931) and difference (𝐴 ∖ 𝐵) (df-dif 3929). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4246 and dfin4 4253. For intersection defined in terms of union, see dfin3 4252. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
Theorem	dfin5 3934*	Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}
Theorem	dfdif2 3935*	Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
Theorem	eldif 3936	Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
Theorem	eldifd 3937	If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3936. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))
Theorem	eldifad 3938	If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3936. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐵)
Theorem	eldifbd 3939	If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3936. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
Theorem	elneeldif 3940	The elements of a set difference and the minuend are not equal. (Contributed by AV, 21-Oct-2023.)
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑋 ≠ 𝑌)
Theorem	velcomp 3941	Characterization of setvar elements of the complement of a class. (Contributed by Andrew Salmon, 15-Jul-2011.)
⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
Theorem	elin 3942	Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2.1.12  Subclasses and subsets
Definition	df-ss 3943*	Define the subclass relationship. Definition 5.9 of [TakeutiZaring] p. 17. For example, {1, 2} ⊆ {1, 2, 3} (ex-ss 30354). Note that 𝐴 ⊆ 𝐴 (proved in ssid 3981). Contrast this relationship with the relationship 𝐴 ⊊ 𝐵 (as will be defined in df-pss 3946). For an alternative definition, not requiring a dummy variable, see dfss2 3944. Other possible definitions are given by dfss3 3947, dfss4 4244, sspss 4077, ssequn1 4161, ssequn2 4164, sseqin2 4198, and ssdif0 4341. 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 11135 and 1ex 11229) or ℝ ( cf. df-r 11137 and reex 11218). 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 11218). This is why we use ⊆ both for subclass relations and for subset relations and call it "subset". (Contributed by NM, 8-Jan-2002.) Revised from the original definition dfss2 3944. (Revised by GG, 15-May-2025.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
Theorem	dfss2 3944	Alternate definition of the subclass relationship between two classes. Exercise 9 of [TakeutiZaring] p. 18. This was the original definition before df-ss 3943. (Contributed by NM, 27-Apr-1994.) Revise df-ss 3943. (Revised by GG, 15-May-2025.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
Theorem	dfss 3945	Variant of subclass definition dfss2 3944. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))
Definition	df-pss 3946	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 30355). Note that ¬ 𝐴 ⊊ 𝐴 (proved in pssirr 4078). Contrast this relationship with the relationship 𝐴 ⊆ 𝐵 (as defined in df-ss 3943). Other possible definitions are given by dfpss2 4063 and dfpss3 4064. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
Theorem	dfss3 3947*	Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	dfss6 3948*	Alternate definition of subclass relationship. (Contributed by RP, 16-Apr-2020.)
⊢ (𝐴 ⊆ 𝐵 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
Theorem	dfssf 3949	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 2376. (Revised by GG, 19-May-2023.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
Theorem	dfss3f 3950	Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	nfss 3951	If 𝑥 is not free in 𝐴 and 𝐵, it is not free in 𝐴 ⊆ 𝐵. (Contributed by NM, 27-Dec-1996.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵
Theorem	ssel 3952	Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.) Avoid ax-12 2177. (Revised by SN, 27-May-2024.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
Theorem	ssel2 3953	Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
Theorem	sseli 3954	Membership implication from subclass relationship. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
Theorem	sselii 3955	Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐶 ∈ 𝐴    ⇒   ⊢ 𝐶 ∈ 𝐵
Theorem	sselid 3956	Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐵)
Theorem	sseld 3957	Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
Theorem	sselda 3958	Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
Theorem	sseldd 3959	Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐵)
Theorem	ssneld 3960	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.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴))
Theorem	ssneldd 3961	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.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
Theorem	ssriv 3962*	Inference based on subclass definition. (Contributed by NM, 21-Jun-1993.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	ssrd 3963	Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	ssrdv 3964*	Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	sstr2 3965	Transitivity of subclass relationship. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Avoid axioms. (Revised by GG, 19-May-2025.)
⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶))
Theorem	sstr2OLD 3966	Obsolete version of sstr2 3965 as of 19-May-2025. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶))
Theorem	sstr 3967	Transitivity of subclass relationship. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)
Theorem	sstri 3968	Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	sstrd 3969	Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sstrid 3970	Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sstrdi 3971	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sylan9ss 3972	A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜓 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)
Theorem	sylan9ssr 3973	A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜓 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶)
Theorem	eqss 3974	The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 21-May-1993.)
⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
Theorem	eqssi 3975	Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐵 ⊆ 𝐴    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqssd 3976	Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	sssseq 3977	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.)
⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 = 𝐵))
Theorem	eqrd 3978	Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.) (Proof shortened by BJ, 1-Dec-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqri 3979	Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqelssd 3980*	Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	ssid 3981	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.)
⊢ 𝐴 ⊆ 𝐴
Theorem	ssidd 3982	Weakening of ssid 3981. (Contributed by BJ, 1-Sep-2022.)
⊢ (𝜑 → 𝐴 ⊆ 𝐴)
Theorem	ssv 3983	Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
⊢ 𝐴 ⊆ V
Theorem	sseq1 3984	Equality theorem for subclasses. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
Theorem	sseq2 3985	Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
Theorem	sseq12 3986	Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷))
Theorem	sseq1i 3987	An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
Theorem	sseq2i 3988	An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)
Theorem	sseq12i 3989	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    ⇒   ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)
Theorem	sseq1d 3990	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
Theorem	sseq2d 3991	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
Theorem	sseq12d 3992	An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷))
Theorem	eqsstrd 3993	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	eqsstrrd 3994	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrd 3995	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrrd 3996	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	eqsstrid 3997	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	eqsstrrid 3998	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrdi 3999	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrrdi 4000	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)