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	3sstr3d 4001	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)
Theorem	3sstr4d 4002	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ⊆ 𝐷)
Theorem	eqimssd 4003	Equality implies inclusion, deduction version. (Contributed by SN, 6-Nov-2024.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	eqimsscd 4004	Equality implies inclusion, deduction version. (Contributed by SN, 15-Feb-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
Theorem	eqimss 4005	Equality implies inclusion. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
Theorem	eqimss2 4006	Equality implies inclusion. (Contributed by NM, 23-Nov-2003.)
⊢ (𝐵 = 𝐴 → 𝐴 ⊆ 𝐵)
Theorem	eqimssi 4007	Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	eqimss2i 4008	Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝐵 ⊆ 𝐴
Theorem	nssne1 4009	Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 ⊆ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	nssne2 4010	Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵)
Theorem	nss 4011*	Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
Theorem	nelss 4012	Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 ⊆ 𝐶)
Theorem	ssrexf 4013	Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))
Theorem	ssrmof 4014	"At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑))
Theorem	ssralv 4015*	Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.) Avoid axioms. (Revised by GG, 19-May-2025.)
⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
Theorem	ssrexv 4016*	Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.) Avoid axioms. (Revised by GG, 19-May-2025.)
⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))
Theorem	ss2ralv 4017*	Two quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023.)
⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑))
Theorem	ss2rexv 4018*	Two existential quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023.)
⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝜑))
Theorem	ssralvOLD 4019*	Obsolete version of ssralv 4015 as of 19-May-2025. (Contributed by NM, 11-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
Theorem	ssrexvOLD 4020*	Obsolete version of ssrexv 4016 as of 19-May-2025. (Contributed by NM, 11-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))
Theorem	ralss 4021*	Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.) Avoid axioms. (Revised by SN, 14-Oct-2025.)
⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑)))
Theorem	rexss 4022*	Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.) Avoid axioms. (Revised by SN, 14-Oct-2025.)
⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝜑)))
Theorem	ralssOLD 4023*	Obsolete version of ralss 4021 as of 14-Oct-2025. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑)))
Theorem	rexssOLD 4024*	Obsolete version of rexss 4022 as of 14-Oct-2025. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝜑)))
Theorem	ss2ab 4025	Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓))
Theorem	abss 4026*	Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴))
Theorem	ssab 4027*	Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Theorem	ssabral 4028*	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 4029*	Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.) (Revised by Steven Nguyen, 28-Jun-2024.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})
Theorem	ss2abi 4030	Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.) Avoid ax-8 2111, ax-10 2142, ax-11 2158, ax-12 2178. (Revised by GG, 28-Jun-2024.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓}
Theorem	abssdv 4031*	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	abssdvOLD 4032*	Obsolete version of abssdv 4031 as of 12-Dec-2024. (Contributed by NM, 20-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)
Theorem	abssi 4033*	Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
⊢ (𝜑 → 𝑥 ∈ 𝐴)    ⇒   ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴
Theorem	ss2rab 4034	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 4035*	Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵))
Theorem	ssrab 4036*	Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	ssrabdv 4037*	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 4038*	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 4039*	Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	ss2rabi 4040	Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabss2 4041*	Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑})
Theorem	ssab2 4042*	Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
Theorem	ssrab2 4043*	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 4044*	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 4045*	Subclass relation for a restricted class abstraction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    ⇒   ⊢ 𝐵 ⊆ 𝐴
Theorem	rabssrabd 4046*	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 4047*	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 4048	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 4049*	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 4050*	A subset relationship useful for converting union to indexed union using dfiun2 4997 or dfiun2g 4994 and intersection to indexed intersection using dfiin2 4998. (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 4051	Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
Theorem	dfpss3 4052	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 4053	Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
Theorem	psseq2 4054	Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))
Theorem	psseq1i 4055	An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)
Theorem	psseq2i 4056	An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)
Theorem	psseq12i 4057	An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)
Theorem	psseq1d 4058	An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
Theorem	psseq2d 4059	An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))
Theorem	psseq12d 4060	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 4061	A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵)
Theorem	pssne 4062	Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵)
Theorem	pssssd 4063	Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	pssned 4064	Proper subclasses are unequal. Deduction form of pssne 4062. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	sspss 4065	Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	pssirr 4066	Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
⊢ ¬ 𝐴 ⊊ 𝐴
Theorem	pssn2lp 4067	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 4068	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 4069	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 4070	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 4071	Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
Theorem	psssstr 4072	Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)
Theorem	psstrd 4073	Proper subclass inclusion is transitive. Deduction form of psstr 4070. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊊ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	sspsstrd 4074	Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4071. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊊ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	psssstrd 4075	Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4072. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	npss 4076	A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3962. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵))
Theorem	ssnelpss 4077	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 4078	Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4077. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐵)
Theorem	ssexnelpss 4079*	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 4080*	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 4081*	Obsolete version of dfdif3 4080 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 4082	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 4083	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 4084	Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))
Theorem	difeq1i 4085	Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
Theorem	difeq2i 4086	Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)
Theorem	difeq12i 4087	Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)
Theorem	difeq1d 4088	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 4089	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 4090	Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))
Theorem	difeqri 4091*	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 4092	Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
Theorem	nfdifOLD 4093	Obsolete version of nfdif 4092 as of 14-May-2025. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
Theorem	eldifi 4094	Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵)
Theorem	eldifn 4095	Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → ¬ 𝐴 ∈ 𝐶)
Theorem	elndif 4096	A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵))
Theorem	neldif 4097	Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶)
Theorem	difdif 4098	Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴
Theorem	difss 4099	Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
Theorem	difssd 4100	A difference of two classes is contained in the minuend. Deduction form of difss 4099. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)