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	ssab 4001*	Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Theorem	ssabral 4002*	The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ss2abdv 4003*	Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 28-Jun-2024.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})
Theorem	ss2abi 4004	Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.) Avoid ax-8 2121, ax-10 2152, ax-11 2168, ax-12 2189. (Revised by GG, 28-Jun-2024.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓}
Theorem	abssdv 4005*	Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.) (Proof shortened by SN, 22-Dec-2024.)
⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)
Theorem	abssi 4006*	Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
⊢ (𝜑 → 𝑥 ∈ 𝐴)    ⇒   ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴
Theorem	ss2rab 4007	Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
Theorem	rabss 4008*	Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵))
Theorem	ssrab 4009*	Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	ss2rabd 4010	Subclass of a restricted class abstraction (deduction form). Saves ax-10 2152, ax-11 2168, ax-12 2189 over using ss2rab 4007 and sylibr 235. (Contributed by SN, 4-Feb-2026.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	ssrabdv 4011*	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 4012*	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 4013*	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 4014	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 4015*	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 4016*	Obsolete version of rabss2 4015 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 4017*	Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
Theorem	ssrab2 4018*	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 4019*	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 4020*	Subclass relation for a restricted class abstraction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    ⇒   ⊢ 𝐵 ⊆ 𝐴
Theorem	rabssrabd 4021*	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 4022*	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 4023	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 4024*	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 4025*	A subset relationship useful for converting union to indexed union using dfiun2 4968 or dfiun2g 4966 and intersection to indexed intersection using dfiin2 4969. (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))
Theorem	dfpss2 4026	Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
Theorem	dfpss3 4027	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 4028	Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
Theorem	psseq2 4029	Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))
Theorem	psseq1i 4030	An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)
Theorem	psseq2i 4031	An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)
Theorem	psseq12i 4032	An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)
Theorem	psseq1d 4033	An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
Theorem	psseq2d 4034	An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))
Theorem	psseq12d 4035	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 4036	A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
Theorem	pssne 4037	Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵)
Theorem	pssssd 4038	Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	pssned 4039	Proper subclasses are unequal. Deduction form of pssne 4037. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	sspss 4040	Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	pssirr 4041	Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
⊢ ¬ 𝐴 ⊊ 𝐴
Theorem	pssn2lp 4042	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 4043	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 4044	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 4045	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 4046	Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
Theorem	psssstr 4047	Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)
Theorem	psstrd 4048	Proper subclass inclusion is transitive. Deduction form of psstr 4045. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊊ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	sspsstrd 4049	Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4046. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊊ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	psssstrd 4050	Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4047. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	npss 4051	A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3937. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵))
Theorem	ssnelpss 4052	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 4053	Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4052. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐵)
Theorem	ssexnelpss 4054*	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)
2.1.13  The difference, union, and intersection of two classes
2.1.13.1  The difference of two classes
Theorem	dfdif3 4055*	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	dfdif3OLD 4056*	Obsolete version of dfdif3 4055 as of 15-Aug-2025. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}
Theorem	difeq1 4057	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 4058	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 4059	Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))
Theorem	difeq1i 4060	Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
Theorem	difeq2i 4061	Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)
Theorem	difeq12i 4062	Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)
Theorem	difeq1d 4063	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 4064	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 4065	Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))
Theorem	difeqri 4066*	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 4067	Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2152, ax-11 2168, ax-12 2189. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
Theorem	eldifi 4068	Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵)
Theorem	eldifn 4069	Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → ¬ 𝐴 ∈ 𝐶)
Theorem	elndif 4070	A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵))
Theorem	neldif 4071	Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶)
Theorem	difdif 4072	Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴
Theorem	difss 4073	Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
Theorem	difssd 4074	A difference of two classes is contained in the minuend. Deduction form of difss 4073. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
Theorem	difss2 4075	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 4076	If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4075. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	ssdifss 4077	Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	ddif 4078	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 4079	Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴)))
Theorem	sscon 4080	Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
Theorem	ssdif 4081	Difference law for subsets. (Contributed by NM, 28-May-1998.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
Theorem	ssdifd 4082	If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 4081. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
Theorem	sscond 4083	If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 4080. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
Theorem	ssdifssd 4084	If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is also contained in 𝐵. Deduction form of ssdifss 4077. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	ssdif2d 4085	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 4086	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 4087	Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.)
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)
Theorem	complss 4088	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 4089	Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.)
⊢ (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵))
2.1.13.2  The union of two classes
Theorem	elun 4090	Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
Theorem	elunnel1 4091	A member of a union that is not a member of the first class, is a member of the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶)
Theorem	elunnel2 4092	A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵)
Theorem	uneqri 4093*	Inference from membership to union. (Contributed by NM, 21-Jun-1993.)
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)    ⇒   ⊢ (𝐴 ∪ 𝐵) = 𝐶
Theorem	unidm 4094	Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 ∪ 𝐴) = 𝐴
Theorem	uncom 4095	Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
Theorem	equncom 4096	If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4096 was automatically derived from equncomVD 45318 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))
Theorem	equncomi 4097	Inference form of equncom 4096. equncomi 4097 was automatically derived from equncomiVD 45319 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
⊢ 𝐴 = (𝐵 ∪ 𝐶)    ⇒   ⊢ 𝐴 = (𝐶 ∪ 𝐵)
Theorem	uneq1 4098	Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.)
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
Theorem	uneq2 4099	Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
Theorem	uneq12 4100	Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))