P. 36 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	csb2 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 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)}
Theorem	csbeq1 3502	Analogue of dfsbcq 3404 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)
Theorem	csbeq2 3503	Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
⊢ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
Theorem	cbvcsb 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.)
⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷
Theorem	cbvcsbv 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 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶
Theorem	csbeq1d 3506	Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)
Theorem	csbid 3507	Analogue of sbid 2100 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ ⦋𝑥 / 𝑥⦌𝐴 = 𝐴
Theorem	csbeq1a 3508	Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
Theorem	csbco 3509*	Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
Theorem	csbtt 3510	Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
Theorem	csbconstgf 3511	Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by NM, 10-Nov-2005.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
Theorem	csbconstg 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.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
Theorem	nfcsb1d 3513	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
Theorem	nfcsb1 3514	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
Theorem	nfcsb1v 3515*	Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
Theorem	nfcsbd 3516	Deduction version of nfcsb 3517. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
Theorem	nfcsb 3517	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
Theorem	csbhypf 3518*	Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3226 for class substitution version. (Contributed by NM, 19-Dec-2008.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbiebt 3519*	Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3523.) (Contributed by NM, 11-Nov-2005.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
Theorem	csbiedf 3520*	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 3521*	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 3522*	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 3523*	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 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    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
Theorem	csbie 3525*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
Theorem	csbied 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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
Theorem	csbied2 3527*	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 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    ⇒   ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷)
Theorem	csbie2 3529*	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 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 = 𝑦 → 𝐵 = 𝐶)    &   ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)
Theorem	cbvralcsf 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.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)
Theorem	cbvrexcsf 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.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)
Theorem	cbvreucsf 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.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓)
Theorem	cbvrabcsf 3534	A more general version of cbvrab 3171 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
Theorem	cbvralv2 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 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)
Theorem	cbvrexv2 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 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)
2.1.11  Define basic set operations and relations
Syntax	cdif 3537	Extend class notation to include class difference (read: "𝐴 minus 𝐵").
class (𝐴 ∖ 𝐵)
Syntax	cun 3538	Extend class notation to include union of two classes (read: "𝐴 union 𝐵").
class (𝐴 ∪ 𝐵)
Syntax	cin 3539	Extend class notation to include the intersection of two classes (read: "𝐴 intersect 𝐵").
class (𝐴 ∩ 𝐵)
Syntax	wss 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 𝐴 ⊆ 𝐵
Syntax	wpss 3541	Extend wff notation with proper subclass relation.
wff 𝐴 ⊊ 𝐵
Theorem	difjust 3542*	Soundness justification theorem for df-dif 3543. (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 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 (𝐴 ∪ 𝐵) (df-un 3545) and intersection (𝐴 ∩ 𝐵) (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 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 3544*	Soundness justification theorem for df-un 3545. (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 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 (𝐴 ∖ 𝐵) (df-dif 3543) and intersection (𝐴 ∩ 𝐵) (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.)
⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
Theorem	injust 3546*	Soundness justification theorem for df-in 3547. (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 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 (𝐴 ∪ 𝐵) (df-un 3545) and difference (𝐴 ∖ 𝐵) (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.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
Theorem	dfin5 3548*	Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}
Theorem	dfdif2 3549*	Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
Theorem	eldif 3550	Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
Theorem	eldifd 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.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))
Theorem	eldifad 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.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐵)
Theorem	eldifbd 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.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
2.1.12  Subclasses and subsets
Definition	df-ss 3554	Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example, {1, 2} ⊆ {1, 2, 3} (ex-ss 26676). Note that 𝐴 ⊆ 𝐴 (proved in ssid 3587). Contrast this relationship with the relationship 𝐴 ⊊ 𝐵 (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.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
Theorem	dfss 3555	Variant of subclass definition df-ss 3554. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))
Definition	df-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 ¬ 𝐴 ⊊ 𝐴 (proved in pssirr 3669). Contrast this relationship with the relationship 𝐴 ⊆ 𝐵 (as defined in df-ss 3554). Other possible definitions are given by dfpss2 3654 and dfpss3 3655. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
Theorem	dfss2 3557*	Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
Theorem	dfss3 3558*	Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	dfss2f 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.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
Theorem	dfss3f 3560	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 3561	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 3562	Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
Theorem	ssel2 3563	Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
Theorem	sseli 3564	Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
Theorem	sselii 3565	Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐶 ∈ 𝐴    ⇒   ⊢ 𝐶 ∈ 𝐵
Theorem	sseldi 3566	Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐵)
Theorem	sseld 3567	Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
Theorem	sselda 3568	Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
Theorem	sseldd 3569	Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐵)
Theorem	ssneld 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.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴))
Theorem	ssneldd 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.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
Theorem	ssriv 3572*	Inference rule based on subclass definition. (Contributed by NM, 21-Jun-1993.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	ssrd 3573	Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	ssrdv 3574*	Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	sstr2 3575	Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶))
Theorem	sstr 3576	Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)
Theorem	sstri 3577	Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	sstrd 3578	Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	syl5ss 3579	Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	syl6ss 3580	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sylan9ss 3581	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 3582	A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜓 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶)
Theorem	eqss 3583	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 3584	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 3585	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	eqrd 3586	Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	ssid 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.)
⊢ 𝐴 ⊆ 𝐴
Theorem	ssv 3588	Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
⊢ 𝐴 ⊆ V
Theorem	sseq1 3589	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 3590	Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
Theorem	sseq12 3591	Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷))
Theorem	sseq1i 3592	An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
Theorem	sseq2i 3593	An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)
Theorem	sseq12i 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    ⇒   ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)
Theorem	sseq1d 3595	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
Theorem	sseq2d 3596	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
Theorem	sseq12d 3597	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	eqsstri 3598	Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	eqsstr3i 3599	Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
⊢ 𝐵 = 𝐴    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	sseqtri 3600	Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶