P. 41 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ssrabdv 4001*	Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)
⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})
Theorem	rabssdv 4002*	Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ 𝐵)
Theorem	ss2rabdv 4003*	Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.) Avoid axioms. (Revised by TM, 1-Feb-2026.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	ss2rabi 4004	Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.) Avoid axioms. (Revised by SN, 4-Feb-2025.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabss2 4005*	Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid axioms. (Revised by TM, 1-Feb-2026.)
⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑})
Theorem	rabss2OLD 4006*	Obsolete version of rabss2 4005 as of 1-Feb-2026. (Contributed by NM, 30-May-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑})
Theorem	ssab2 4007*	Subclass relation for the restriction of a class abstraction. Prefer using the more natural statement ssrab2 4008. (Contributed by NM, 31-Mar-1995.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
Theorem	ssrab2 4008*	Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.) (Proof shortened by BJ and SN, 8-Aug-2024.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
Theorem	rabss3d 4009*	Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	ssrab3 4010*	Subclass relation for a restricted class abstraction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    ⇒   ⊢ 𝐵 ⊆ 𝐴
Theorem	rabssrabd 4011*	Subclass of a restricted class abstraction. (Contributed by AV, 4-Jun-2022.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	ssrabeq 4012*	If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})
Theorem	rabssab 4013	A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
Theorem	eqrrabd 4014*	Deduce equality with a restricted abstraction. (Contributed by Thierry Arnoux, 11-Apr-2024.)
⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓})
Theorem	uniiunlem 4015*	A subset relationship useful for converting union to indexed union using dfiun2 4993 or dfiun2g 4991 and intersection to indexed intersection using dfiin2 4994. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐷 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶))
2.1.11.2  Proper subclasses and subsets
Syntax	wpss 4016	Syntax for the proper subclass predicate.
wff 𝐴 ⊊ 𝐵
Definition	df-pss 4017	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 31062). Note that ¬ 𝐴 ⊊ 𝐴 (proved in pssirr 4033). Contrast this relationship with the relationship 𝐴 ⊆ 𝐵 (as defined in df-ss 3901). Other possible definitions are given by dfpss2 4018 and dfpss3 4019. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
Theorem	dfpss2 4018	Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
Theorem	dfpss3 4019	Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
Theorem	psseq1 4020	Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
Theorem	psseq2 4021	Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))
Theorem	psseq1i 4022	An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)
Theorem	psseq2i 4023	An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)
Theorem	psseq12i 4024	An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)
Theorem	psseq1d 4025	An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
Theorem	psseq2d 4026	An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))
Theorem	psseq12d 4027	An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))
Theorem	pssss 4028	A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
Theorem	pssne 4029	Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵)
Theorem	pssssd 4030	Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	pssned 4031	Proper subclasses are unequal. Deduction form of pssne 4029. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	sspss 4032	Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	pssirr 4033	Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ¬ 𝐴 ⊊ 𝐴
Theorem	pssirrOLD 4034	Obsolete version of pssirr 4033 as of 10-Jun-2026. (Contributed by NM, 7-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝐴 ⊊ 𝐴
Theorem	pssn2lp 4035	Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴)
Theorem	sspsstri 4036	Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
Theorem	ssnpss 4037	Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴)
Theorem	psstr 4038	Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
Theorem	sspsstr 4039	Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
Theorem	psssstr 4040	Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)
Theorem	psstrd 4041	Proper subclass inclusion is transitive. Deduction form of psstr 4038. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊊ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	sspsstrd 4042	Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4039. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊊ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	psssstrd 4043	Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4040. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	npss 4044	A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3928. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵))
Theorem	ssnelpss 4045	A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵))
Theorem	ssnelpssd 4046	Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4045. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐵)
Theorem	ssexnelpss 4047*	If there is an element of a class which is not contained in a subclass, the subclass is a proper subclass. (Contributed by AV, 29-Jan-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵)
2.1.12  Basic operations on classes
2.1.12.1  The difference of two classes
Syntax	cdif 4048	Syntax for the difference of two classes.
class (𝐴 ∖ 𝐵)
Theorem	difjust 4049*	Soundness justification theorem for df-dif 4050. (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 4050*	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ({1, 3} ∖ {1, 8}) = {3} (ex-dif 31057). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 4154) and intersection (𝐴 ∩ 𝐵) (df-in 4094). 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	dfdif2 4051*	Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
Theorem	eldif 4052	Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
Theorem	eldifd 4053	If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 4052. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))
Theorem	eldifad 4054	If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 4052. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐵)
Theorem	eldifbd 4055	If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 4052. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
Theorem	elneeldif 4056	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 4057	Characterization of setvar elements of the complement of a class. (Contributed by Andrew Salmon, 15-Jul-2011.)
⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
Theorem	dfdif3 4058*	Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.) (Proof shortened by SN, 15-Aug-2025.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}
Theorem	difeq1 4059	Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
Theorem	difeq2 4060	Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))
Theorem	difeq12 4061	Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))
Theorem	difeq1i 4062	Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
Theorem	difeq2i 4063	Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)
Theorem	difeq12i 4064	Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)
Theorem	difeq1d 4065	Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
Theorem	difeq2d 4066	Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))
Theorem	difeq12d 4067	Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))
Theorem	difeqri 4068*	Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)    ⇒   ⊢ (𝐴 ∖ 𝐵) = 𝐶
Theorem	nfdif 4069	Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2180, ax-11 2197, ax-12 2216. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
Theorem	eldifi 4070	Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵)
Theorem	eldifn 4071	Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → ¬ 𝐴 ∈ 𝐶)
Theorem	elndif 4072	A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵))
Theorem	neldif 4073	Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶)
Theorem	difdif 4074	Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴
Theorem	difss 4075	Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
Theorem	difssd 4076	A difference of two classes is contained in the minuend. Deduction form of difss 4075. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
Theorem	difss2 4077	If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)
Theorem	difss2d 4078	If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4077. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	ssdifss 4079	Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	ddif 4080	Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
⊢ (V ∖ (V ∖ 𝐴)) = 𝐴
Theorem	ssconb 4081	Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴)))
Theorem	sscon 4082	Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
Theorem	ssdif 4083	Difference law for subsets. (Contributed by NM, 28-May-1998.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
Theorem	ssdifd 4084	If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 4083. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
Theorem	sscond 4085	If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 4082. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
Theorem	ssdifssd 4086	If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is also contained in 𝐵. Deduction form of ssdifss 4079. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	ssdif2d 4087	If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴 ∖ 𝐷) is contained in (𝐵 ∖ 𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶))
Theorem	raldifb 4088	Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)
Theorem	rexdifi 4089	Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.)
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)
Theorem	complss 4090	Complementation reverses inclusion. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 19-Mar-2021.)
⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))
Theorem	compleq 4091	Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.)
⊢ (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵))
2.1.12.2  The intersection of two classes
Syntax	cin 4092	Syntax for the intersection of two classes.
class (𝐴 ∩ 𝐵)
Theorem	injust 4093*	Soundness justification theorem for df-in 4094. (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 4094*	Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 31059). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 4154) and difference (𝐴 ∖ 𝐵) (df-dif 4050). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4219 and dfin4 4226. For intersection defined in terms of union, see dfin3 4225. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
Theorem	dfin5 4095*	Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}
Theorem	elin 4096	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))
Theorem	dfss2 4097	Alternate definition of the subclass relationship between two classes. Exercise 9 of [TakeutiZaring] p. 18. This was the original definition before df-ss 3901. (Contributed by NM, 27-Apr-1994.) Revise df-ss 3901. (Revised by GG, 15-May-2025.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
Theorem	dfss 4098	Variant of subclass definition dfss2 4097. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))
Theorem	elini 4099	Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)
Theorem	elind 4100	Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))