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Theorem List for Intuitionistic Logic Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 3001 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that 𝐴𝐴 (proved in ssid 3033). For a more traditional definition, but requiring a dummy variable, see dfss2 3003. Other possible definitions are given by dfss3 3004, ssequn1 3159, ssequn2 3162, and sseqin2 3208. (Contributed by NM, 27-Apr-1994.)
(𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
 
Theoremdfss 3002 Variant of subclass definition df-ss 3001. (Contributed by NM, 3-Sep-2004.)
(𝐴𝐵𝐴 = (𝐴𝐵))
 
Theoremdfss2 3003* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
(𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremdfss3 3004* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremdfss2f 3005 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.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremdfss3f 3006 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremnfss 3007 If 𝑥 is not free in 𝐴 and 𝐵, it is not free in 𝐴𝐵. (Contributed by NM, 27-Dec-1996.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremssel 3008 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremssel2 3009 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
((𝐴𝐵𝐶𝐴) → 𝐶𝐵)
 
Theoremsseli 3010 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremsselii 3011 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
𝐴𝐵    &   𝐶𝐴       𝐶𝐵
 
Theoremsseldi 3012 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
𝐴𝐵    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)
 
Theoremsseld 3013 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremsselda 3014 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
(𝜑𝐴𝐵)       ((𝜑𝐶𝐴) → 𝐶𝐵)
 
Theoremsseldd 3015 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)
 
Theoremssneld 3016 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.)
(𝜑𝐴𝐵)       (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))
 
Theoremssneldd 3017 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.)
(𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐶𝐵)       (𝜑 → ¬ 𝐶𝐴)
 
Theoremssriv 3018* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)
(𝑥𝐴𝑥𝐵)       𝐴𝐵
 
Theoremssrd 3019 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
 
Theoremssrdv 3020* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
(𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
 
Theoremsstr2 3021 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝐴𝐵 → (𝐵𝐶𝐴𝐶))
 
Theoremsstr 3022 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsstri 3023 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
𝐴𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremsstrd 3024 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl5ss 3025 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
𝐴𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl6ss 3026 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsylan9ss 3027 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜑𝜓) → 𝐴𝐶)
 
Theoremsylan9ssr 3028 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜓𝜑) → 𝐴𝐶)
 
Theoremeqss 3029 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
 
Theoremeqssi 3030 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
𝐴𝐵    &   𝐵𝐴       𝐴 = 𝐵
 
Theoremeqssd 3031 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = 𝐵)
 
Theoremeqrd 3032 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremssid 3033 Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
𝐴𝐴
 
Theoremssidd 3034 Weakening of ssid 3033. (Contributed by BJ, 1-Sep-2022.)
(𝜑𝐴𝐴)
 
Theoremssv 3035 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
𝐴 ⊆ V
 
Theoremsseq1 3036 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremsseq2 3037 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremsseq12 3038 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
 
Theoremsseq1i 3039 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremsseq2i 3040 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremsseq12i 3041 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       (𝐴𝐶𝐵𝐷)
 
Theoremsseq1d 3042 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremsseq2d 3043 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremsseq12d 3044 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremeqsstri 3045 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqsstr3i 3046 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
𝐵 = 𝐴    &   𝐵𝐶       𝐴𝐶
 
Theoremsseqtri 3047 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
 
Theoremsseqtr4i 3048 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶
 
Theoremeqsstrd 3049 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqsstr3d 3050 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐵 = 𝐴)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsseqtrd 3051 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremsseqtr4d 3052 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theorem3sstr3i 3053 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝐷
 
Theorem3sstr4i 3054 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷
 
Theorem3sstr3g 3055 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)
 
Theorem3sstr4g 3056 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)
 
Theorem3sstr3d 3057 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)
 
Theorem3sstr4d 3058 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)
 
Theoremsyl5eqss 3059 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
𝐴 = 𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl5eqssr 3060 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl6sseq 3061 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6sseqr 3062 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝐶)
 
Theoremsyl5sseq 3063 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐵𝐴    &   (𝜑𝐴 = 𝐶)       (𝜑𝐵𝐶)
 
Theoremsyl5sseqr 3064 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐵𝐴    &   (𝜑𝐶 = 𝐴)       (𝜑𝐵𝐶)
 
Theoremsyl6eqss 3065 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6eqssr 3066 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐵 = 𝐴)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremeqimss 3067 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵𝐴𝐵)
 
Theoremeqimss2 3068 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
(𝐵 = 𝐴𝐴𝐵)
 
Theoremeqimssi 3069 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
𝐴 = 𝐵       𝐴𝐵
 
Theoremeqimss2i 3070 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
𝐴 = 𝐵       𝐵𝐴
 
Theoremnssne1 3071 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
 
Theoremnssne2 3072 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
 
Theoremnssr 3073* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
(∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ 𝐴𝐵)
 
Theoremssralv 3074* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
(𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))
 
Theoremssrexv 3075* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
(𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
 
Theoremralss 3076* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 (𝑥𝐴𝜑)))
 
Theoremrexss 3077* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 (𝑥𝐴𝜑)))
 
Theoremss2ab 3078 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
 
Theoremabss 3079* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
 
Theoremssab 3080* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
(𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theoremssabral 3081* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
(𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥𝐴 𝜑)
 
Theoremss2abi 3082 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
(𝜑𝜓)       {𝑥𝜑} ⊆ {𝑥𝜓}
 
Theoremss2abdv 3083* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
 
Theoremabssdv 3084* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} ⊆ 𝐴)
 
Theoremabssi 3085* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑𝑥𝐴)       {𝑥𝜑} ⊆ 𝐴
 
Theoremss2rab 3086 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))
 
Theoremrabss 3087* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))
 
Theoremssrab 3088* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
(𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
 
Theoremssrabdv 3089* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐵) → 𝜓)       (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
 
Theoremrabssdv 3090* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
((𝜑𝑥𝐴𝜓) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ 𝐵)
 
Theoremss2rabdv 3091* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
 
Theoremss2rabi 3092 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}
 
Theoremrabss2 3093* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → {𝑥𝐴𝜑} ⊆ {𝑥𝐵𝜑})
 
Theoremssab2 3094* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
{𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
 
Theoremssrab2 3095* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
{𝑥𝐴𝜑} ⊆ 𝐴
 
Theoremssrabeq 3096* 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 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
 
Theoremrabssab 3097 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
{𝑥𝐴𝜑} ⊆ {𝑥𝜑}
 
Theoremuniiunlem 3098* A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
(∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶))
 
2.1.13  The difference, union, and intersection of two classes
 
2.1.13.1  The difference of two classes
 
Theoremdfdif3 3099* Alternate definition of class difference. Definition of relative set complement in Section 2.3 of [Pierik], p. 10. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.)
(𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
 
Theoremdifeq1 3100 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
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