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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfss2 3901* | 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 2142, ax-11 2158, ax-12 2175. (Revised by SN, 16-May-2024.) |
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Theorem | dfss2OLD 3902* | Obsolete version of dfss2 3901 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 3903* | Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.) |
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) | ||
Theorem | dfss6 3904* | Alternate definition of subclass relationship. (Contributed by RP, 16-Apr-2020.) |
⊢ (𝐴 ⊆ 𝐵 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | ||
Theorem | dfss2f 3905 | 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 2379. (Revised by Gino Giotto, 19-May-2023.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Theorem | dfss3f 3906 | 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 3907 | 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 3908 | Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.) Avoid ax-12 2175. (Revised by SN, 27-May-2024.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | ||
Theorem | sselOLD 3909 | Obsolete version of ssel 3908 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 3910 | Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵) | ||
Theorem | sseli 3911 | Membership implication from subclass relationship. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) | ||
Theorem | sselii 3912 | Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ 𝐶 ∈ 𝐴 ⇒ ⊢ 𝐶 ∈ 𝐵 | ||
Theorem | sseldi 3913 | Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐵) | ||
Theorem | sseld 3914 | Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | ||
Theorem | sselda 3915 | Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵) | ||
Theorem | sseldd 3916 | Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐶 ∈ 𝐵) | ||
Theorem | ssneld 3917 | 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 3918 | 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 3919* | Inference based on subclass definition. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ⇒ ⊢ 𝐴 ⊆ 𝐵 | ||
Theorem | ssrd 3920 | 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 3921* | Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | sstr2 3922 | 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 3923 | Transitivity of subclass relationship. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | ||
Theorem | sstri 3924 | Subclass transitivity inference. (Contributed by NM, 5-May-2000.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ 𝐴 ⊆ 𝐶 | ||
Theorem | sstrd 3925 | Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | sstrid 3926 | Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | sstrdi 3927 | Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | sylan9ss 3928 | 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 3929 | A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜓 → 𝐵 ⊆ 𝐶) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶) | ||
Theorem | eqss 3930 | 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 3931 | 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 3932 | 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 3933 | 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 3934 | 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 3935 | 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 3936* | 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 3937 | 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 3938 | Weakening of ssid 3937. (Contributed by BJ, 1-Sep-2022.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐴) | ||
Theorem | ssv 3939 | Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.) |
⊢ 𝐴 ⊆ V | ||
Theorem | sseq1 3940 | 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 3941 | Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.) |
⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)) | ||
Theorem | sseq12 3942 | Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) | ||
Theorem | sseq1i 3943 | An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶) | ||
Theorem | sseq2i 3944 | An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵) | ||
Theorem | sseq12i 3945 | 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 3946 | An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | ||
Theorem | sseq2d 3947 | An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)) | ||
Theorem | sseq12d 3948 | 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 3949 | Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ 𝐴 ⊆ 𝐶 | ||
Theorem | eqsstrri 3950 | Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.) |
⊢ 𝐵 = 𝐴 & ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ 𝐴 ⊆ 𝐶 | ||
Theorem | sseqtri 3951 | Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴 ⊆ 𝐶 | ||
Theorem | sseqtrri 3952 | Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴 ⊆ 𝐶 | ||
Theorem | eqsstrd 3953 | Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | eqsstrrd 3954 | Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | sseqtrd 3955 | Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | sseqtrrd 3956 | Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | 3sstr3i 3957 | 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 3958 | 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 3959 | 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 3960 | 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 3961 | 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 3962 | 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 3963 | A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | eqsstrrid 3964 | A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | sseqtrdi 3965 | A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | sseqtrrdi 3966 | A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | sseqtrid 3967 | Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐵 ⊆ 𝐴 & ⊢ (𝜑 → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | sseqtrrid 3968 | Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐵 ⊆ 𝐴 & ⊢ (𝜑 → 𝐶 = 𝐴) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | eqsstrdi 3969 | A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | eqsstrrdi 3970 | A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | eqimss 3971 | Equality implies the subclass relation. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | ||
Theorem | eqimss2 3972 | Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.) |
⊢ (𝐵 = 𝐴 → 𝐴 ⊆ 𝐵) | ||
Theorem | eqimssi 3973 | Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝐴 ⊆ 𝐵 | ||
Theorem | eqimss2i 3974 | Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝐵 ⊆ 𝐴 | ||
Theorem | nssne1 3975 | 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 3976 | 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 3977* | 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 3978 | 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 3979 | 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 3980 | "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 3981* | Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.) |
⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | ssrexv 3982* | Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.) |
⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | ss2ralv 3983* | 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 3984* | 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 3985* | 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 3986* | 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 3987 | Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.) |
⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | abss 3988* | Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.) |
⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) | ||
Theorem | ssab 3989* | Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.) |
⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | ||
Theorem | ssabral 3990* | 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 3991* | Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.) Avoid ax-8 2113, ax-10 2142, ax-11 2158, ax-12 2175. (Revised by Gino Giotto, 28-Jun-2024.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) | ||
Theorem | ss2abdvALT 3992* | Alternate proof of ss2abdv 3991. Shorter, but requiring ax-8 2113. (Contributed by Steven Nguyen, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) | ||
Theorem | ss2abdvOLD 3993* | Obsolete version of ss2abdv 3991 as of 28-Jun-2024. (Contributed by NM, 29-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) | ||
Theorem | ss2abi 3994 | Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.) Avoid ax-8 2113, ax-10 2142, ax-11 2158, ax-12 2175. (Revised by Gino Giotto, 28-Jun-2024.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} | ||
Theorem | ss2abiOLD 3995 | Obsolete version of ss2abi 3994 as of 28-Jun-2024. (Contributed by NM, 31-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} | ||
Theorem | abssdv 3996* | Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.) |
⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴) | ||
Theorem | abssi 3997* | Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.) |
⊢ (𝜑 → 𝑥 ∈ 𝐴) ⇒ ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴 | ||
Theorem | ss2rab 3998 | 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 3999* | Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) | ||
Theorem | ssrab 4000* | Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.) |
⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑)) |
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