P. 40 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eldif 3901	Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
Theorem	eldifd 3902	If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3901. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))
Theorem	eldifad 3903	If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3901. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐵)
Theorem	eldifbd 3904	If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3901. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
Theorem	elneeldif 3905	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 3906	Characterization of setvar elements of the complement of a class. (Contributed by Andrew Salmon, 15-Jul-2011.)
⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
Theorem	elin 3907	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))
2.1.12  Subclasses and subsets
Definition	df-ss 3908	Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example, {1, 2} ⊆ {1, 2, 3} (ex-ss 28770). Note that 𝐴 ⊆ 𝐴 (proved in ssid 3947). Contrast this relationship with the relationship 𝐴 ⊊ 𝐵 (as will be defined in df-pss 3910). For a more traditional definition, but requiring a dummy variable, see dfss2 3911. Other possible definitions are given by dfss3 3913, dfss4 4197, sspss 4038, ssequn1 4118, ssequn2 4121, sseqin2 4154, and ssdif0 4302. We prefer the label "ss" ("subset") for ⊆, despite the fact that it applies to classes. It is much more common to refer to this as the subset relation than subclass, especially since most of the time the arguments are in fact sets (and for pragmatic reasons we don't want to need to use different operations for sets). The way set.mm is set up, many things are technically classes despite morally (and provably) being sets, like 1 (cf. df-1 10863 and 1ex 10955) or ℝ ( cf. df-r 10865 and reex 10946). This has to do with the fact that there are no "set expressions": classes are expressions but there are only set variables in set.mm (cf. https://us.metamath.org/downloads/grammar-ambiguity.txt 10946). This is why we use ⊆ both for subclass relations and for subset relations and call it "subset". (Contributed by NM, 27-Apr-1994.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
Theorem	dfss 3909	Variant of subclass definition df-ss 3908. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵))
Definition	df-pss 3910	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 28771). Note that ¬ 𝐴 ⊊ 𝐴 (proved in pssirr 4039). Contrast this relationship with the relationship 𝐴 ⊆ 𝐵 (as defined in df-ss 3908). Other possible definitions are given by dfpss2 4024 and dfpss3 4025. (Contributed by NM, 7-Feb-1996.)
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
Theorem	dfss2 3911*	Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.) Avoid ax-10 2140, ax-11 2157, ax-12 2174. (Revised by SN, 16-May-2024.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
Theorem	dfss2OLD 3912*	Obsolete version of dfss2 3911 as of 16-May-2024. (Contributed by NM, 8-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
Theorem	dfss3 3913*	Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	dfss6 3914*	Alternate definition of subclass relationship. (Contributed by RP, 16-Apr-2020.)
⊢ (𝐴 ⊆ 𝐵 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
Theorem	dfss2f 3915	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.) Avoid ax-13 2373. (Revised by Gino Giotto, 19-May-2023.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
Theorem	dfss3f 3916	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 3917	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 3918	Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.) Avoid ax-12 2174. (Revised by SN, 27-May-2024.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
Theorem	sselOLD 3919	Obsolete version of ssel 3918 as of 27-May-2024. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
Theorem	ssel2 3920	Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
Theorem	sseli 3921	Membership implication from subclass relationship. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
Theorem	sselii 3922	Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐶 ∈ 𝐴    ⇒   ⊢ 𝐶 ∈ 𝐵
Theorem	sselid 3923	Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐵)
Theorem	sseld 3924	Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
Theorem	sselda 3925	Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
Theorem	sseldd 3926	Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐵)
Theorem	ssneld 3927	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 3928	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 3929*	Inference based on subclass definition. (Contributed by NM, 21-Jun-1993.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	ssrd 3930	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 3931*	Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	sstr2 3932	Transitivity of subclass relationship. 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 3933	Transitivity of subclass relationship. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)
Theorem	sstri 3934	Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	sstrd 3935	Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sstrid 3936	Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sstrdi 3937	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sylan9ss 3938	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 3939	A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜓 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶)
Theorem	eqss 3940	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 3941	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 3942	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	sssseq 3943	If a class is a subclass of another class, then the classes are equal if and only if the other class is a subclass of the first class. (Contributed by AV, 23-Dec-2020.)
⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 = 𝐵))
Theorem	eqrd 3944	Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.) (Proof shortened by BJ, 1-Dec-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqri 3945	Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqelssd 3946*	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 3947	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	ssidd 3948	Weakening of ssid 3947. (Contributed by BJ, 1-Sep-2022.)
⊢ (𝜑 → 𝐴 ⊆ 𝐴)
Theorem	ssv 3949	Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
⊢ 𝐴 ⊆ V
Theorem	sseq1 3950	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 3951	Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
Theorem	sseq12 3952	Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷))
Theorem	sseq1i 3953	An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
Theorem	sseq2i 3954	An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)
Theorem	sseq12i 3955	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 3956	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
Theorem	sseq2d 3957	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵))
Theorem	sseq12d 3958	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 3959	Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	eqsstrri 3960	Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
⊢ 𝐵 = 𝐴    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	sseqtri 3961	Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	sseqtrri 3962	Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ⊆ 𝐶
Theorem	eqsstrd 3963	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	eqsstrrd 3964	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrd 3965	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrrd 3966	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	3sstr3i 3967	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 3968	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 3969	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 3970	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 3971	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 3972	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	eqsstrid 3973	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	eqsstrrid 3974	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrdi 3975	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrrdi 3976	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	sseqtrid 3977	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐵 ⊆ 𝐴    &   ⊢ (𝜑 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
Theorem	sseqtrrid 3978	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐵 ⊆ 𝐴    &   ⊢ (𝜑 → 𝐶 = 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
Theorem	eqsstrdi 3979	A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	eqsstrrdi 3980	A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	eqimss 3981	Equality implies inclusion. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
Theorem	eqimss2 3982	Equality implies inclusion. (Contributed by NM, 23-Nov-2003.)
⊢ (𝐵 = 𝐴 → 𝐴 ⊆ 𝐵)
Theorem	eqimssi 3983	Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	eqimss2i 3984	Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝐵 ⊆ 𝐴
Theorem	nssne1 3985	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 3986	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 3987*	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 3988	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 3989	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 3990	"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 3991*	Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
Theorem	ssrexv 3992*	Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))
Theorem	ss2ralv 3993*	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 3994*	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	ralss 3995*	Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑)))
Theorem	rexss 3996*	Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝜑)))
Theorem	ss2ab 3997	Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓))
Theorem	abss 3998*	Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴))
Theorem	ssab 3999*	Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Theorem	ssabral 4000*	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)