P. 32 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	csbiegf 3101*	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 3102*	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 3103*	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 3104*	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 3105*	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 3106*	Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3107). (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 3107*	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 3108*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 2998 avoids a disjointness condition on 𝑥 and 𝐴 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	sbcnestgf 3109	Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
Theorem	csbnestgf 3110	Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
Theorem	sbcnestg 3111*	Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
Theorem	csbnestg 3112*	Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
Theorem	csbnest1g 3113	Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶)
Theorem	csbidmg 3114*	Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
Theorem	sbcco3g 3115*	Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑))
Theorem	csbco3g 3116*	Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷)
Theorem	rspcsbela 3117*	Special case related to rspsbc 3046. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷)
Theorem	sbnfc2 3118*	Two ways of expressing "𝑥 is (effectively) not free in 𝐴." (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
Theorem	csbabg 3119*	Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑})
Theorem	cbvralcsf 3120	A more general version of cbvralf 2697 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 3121	A more general version of cbvrexf 2698 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 3122	A more general version of cbvreuv 2706 that has no distinct variable rextrictions. 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 3123	A more general version of cbvrab 2736 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 3124*	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 3125*	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)
Theorem	rspc2vd 3126*	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 3127	Extend class notation to include class difference (read: "𝐴 minus 𝐵").
class (𝐴 ∖ 𝐵)
Syntax	cun 3128	Extend class notation to include union of two classes (read: "𝐴 union 𝐵").
class (𝐴 ∪ 𝐵)
Syntax	cin 3129	Extend class notation to include the intersection of two classes (read: "𝐴 intersect 𝐵").
class (𝐴 ∩ 𝐵)
Syntax	wss 3130	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
Theorem	difjust 3131*	Soundness justification theorem for df-dif 3132. (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 3132*	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3134) and intersection (𝐴 ∩ 𝐵) (df-in 3136). 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 3133*	Soundness justification theorem for df-un 3134. (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 3134*	Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference (𝐴 ∖ 𝐵) (df-dif 3132) and intersection (𝐴 ∩ 𝐵) (df-in 3136). (Contributed by NM, 23-Aug-1993.)
⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
Theorem	injust 3135*	Soundness justification theorem for df-in 3136. (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 3136*	Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3134) and difference (𝐴 ∖ 𝐵) (df-dif 3132). (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
Theorem	dfin5 3137*	Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}
Theorem	dfdif2 3138*	Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
Theorem	eldif 3139	Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
Theorem	eldifd 3140	If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3139. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))
Theorem	eldifad 3141	If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3139. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐵)
Theorem	eldifbd 3142	If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3139. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
2.1.12  Subclasses and subsets
Definition	df-ss 3143	Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that 𝐴 ⊆ 𝐴 (proved in ssid 3176). For a more traditional definition, but requiring a dummy variable, see dfss2 3145. Other possible definitions are given by dfss3 3146, ssequn1 3306, ssequn2 3309, and sseqin2 3355. (Contributed by NM, 27-Apr-1994.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
Theorem	dfss 3144	Variant of subclass definition df-ss 3143. (Contributed by NM, 3-Sep-2004.)
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))
Theorem	dfss2 3145*	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 3146*	Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	dfss2f 3147	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 3148	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 3149	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 3150	Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
Theorem	ssel2 3151	Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
Theorem	sseli 3152	Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
Theorem	sselii 3153	Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐶 ∈ 𝐴    ⇒   ⊢ 𝐶 ∈ 𝐵
Theorem	sselid 3154	Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐵)
Theorem	sseld 3155	Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
Theorem	sselda 3156	Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
Theorem	sseldd 3157	Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐵)
Theorem	ssneld 3158	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 3159	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 3160*	Inference based on subclass definition. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	ssrd 3161	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 3162*	Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	sstr2 3163	Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶))
Theorem	sstr 3164	Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)
Theorem	sstri 3165	Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	sstrd 3166	Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sstrid 3167	Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sstrdi 3168	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sylan9ss 3169	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 3170	A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜓 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶)
Theorem	eqss 3171	The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
Theorem	eqssi 3172	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 3173	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 3174	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	eqelssd 3175*	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 3176	Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ 𝐴 ⊆ 𝐴
Theorem	ssidd 3177	Weakening of ssid 3176. (Contributed by BJ, 1-Sep-2022.)
⊢ (𝜑 → 𝐴 ⊆ 𝐴)
Theorem	ssv 3178	Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
⊢ 𝐴 ⊆ V
Theorem	sseq1 3179	Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
Theorem	sseq2 3180	Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
Theorem	sseq12 3181	Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷))
Theorem	sseq1i 3182	An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
Theorem	sseq2i 3183	An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)
Theorem	sseq12i 3184	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 3185	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
Theorem	sseq2d 3186	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
Theorem	sseq12d 3187	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 3188	Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	eqsstrri 3189	Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
⊢ 𝐵 = 𝐴    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	sseqtri 3190	Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	sseqtrri 3191	Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	eqsstrd 3192	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	eqsstrrd 3193	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrd 3194	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrrd 3195	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	3sstr3i 3196	Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶 ⊆ 𝐷
Theorem	3sstr4i 3197	Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶 ⊆ 𝐷
Theorem	3sstr3g 3198	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶 ⊆ 𝐷)
Theorem	3sstr4g 3199	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 ⊆ 𝐷)
Theorem	3sstr3d 3200	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 ⊆ 𝐷)