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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sbcexf 38101* | Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) | ||
Theorem | sbcalfi 38102* | Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
⊢ Ⅎ𝑦𝐴 & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | sbcexfi 38103* | Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
⊢ Ⅎ𝑦𝐴 & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓) | ||
Theorem | spsbcdi 38104 | A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
⊢ 𝐴 ∈ V & ⊢ (𝜑 → ∀𝑥𝜒) & ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | alrimii 38105* | A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝜓) & ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓) & ⊢ Ⅎ𝑦𝜒 ⇒ ⊢ (𝜑 → ∀𝑥𝜒) | ||
Theorem | spesbcdi 38106 | A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
⊢ (𝜑 → 𝜓) & ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜒) | ||
Theorem | exlimddvf 38107 | A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.) |
⊢ (𝜑 → ∃𝑥𝜃) & ⊢ Ⅎ𝑥𝜓 & ⊢ ((𝜃 ∧ 𝜓) → 𝜒) & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | exlimddvfi 38108 | A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.) |
⊢ (𝜑 → ∃𝑥𝜃) & ⊢ Ⅎ𝑦𝜃 & ⊢ Ⅎ𝑦𝜓 & ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂) & ⊢ ((𝜂 ∧ 𝜓) → 𝜒) & ⊢ Ⅎ𝑦𝜒 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | sbceq1ddi 38109 | A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝜃) & ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜃) & ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜂) | ||
Theorem | sbccom2lem 38110* | Lemma for sbccom2 38111. (Contributed by Giovanni Mascellani, 31-May-2019.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | ||
Theorem | sbccom2 38111* | Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | ||
Theorem | sbccom2f 38112* | Commutative law for double class substitution, with nonfree variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.) |
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | ||
Theorem | sbccom2fi 38113* | Commutative law for double class substitution, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.) |
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝐴 & ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓) | ||
Theorem | csbcom2fi 38114* | Commutative law for double class substitution in a class, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.) |
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝐴 & ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 & ⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸 ⇒ ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸 | ||
A collection of Tseitin axioms used to convert a wff to Conjunctive Normal Form. | ||
Theorem | fald 38115 | Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ¬ ⊥) | ||
Theorem | tsim1 38116 | A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓))) | ||
Theorem | tsim2 38117 | A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜑 ∨ (𝜑 → 𝜓))) | ||
Theorem | tsim3 38118 | A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 → 𝜓))) | ||
Theorem | tsbi1 38119 | A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓))) | ||
Theorem | tsbi2 38120 | A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓))) | ||
Theorem | tsbi3 38121 | A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓))) | ||
Theorem | tsbi4 38122 | A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓))) | ||
Theorem | tsxo1 38123 | A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓))) | ||
Theorem | tsxo2 38124 | A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓))) | ||
Theorem | tsxo3 38125 | A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ⊻ 𝜓))) | ||
Theorem | tsxo4 38126 | A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ (𝜑 ⊻ 𝜓))) | ||
Theorem | tsan1 38127 | A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓))) | ||
Theorem | tsan2 38128 | A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜑 ∨ ¬ (𝜑 ∧ 𝜓))) | ||
Theorem | tsan3 38129 | A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓))) | ||
Theorem | tsna1 38130 | A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓))) | ||
Theorem | tsna2 38131 | A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜑 ∨ (𝜑 ⊼ 𝜓))) | ||
Theorem | tsna3 38132 | A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜓 ∨ (𝜑 ⊼ 𝜓))) | ||
Theorem | tsor1 38133 | A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓))) | ||
Theorem | tsor2 38134 | A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → (¬ 𝜑 ∨ (𝜑 ∨ 𝜓))) | ||
Theorem | tsor3 38135 | A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 ∨ 𝜓))) | ||
Theorem | ts3an1 38136 | A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ∨ (𝜑 ∧ 𝜓 ∧ 𝜒))) | ||
Theorem | ts3an2 38137 | A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))) | ||
Theorem | ts3an3 38138 | A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → (𝜒 ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))) | ||
Theorem | ts3or1 38139 | A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → (((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ 𝜓 ∨ 𝜒))) | ||
Theorem | ts3or2 38140 | A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜓 ∨ 𝜒))) | ||
Theorem | ts3or3 38141 | A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → (¬ 𝜒 ∨ (𝜑 ∨ 𝜓 ∨ 𝜒))) | ||
A collection of theorems for commuting equalities (or biconditionals) with other constructs. | ||
Theorem | iuneq2f 38142 | Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | rabeq12f 38143 | Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓}) | ||
Theorem | csbeq12 38144 | Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷) | ||
Theorem | sbeqi 38145 | Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ ((𝑥 = 𝑦 ∧ ∀𝑧(𝜑 ↔ 𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓)) | ||
Theorem | ralbi12f 38146 | Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | oprabbi 38147 | Equality deduction for class abstraction of nested ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) | ||
Theorem | mpobi123f 38148* | Equality deduction for maps-to notations with two arguments. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑦𝐶 & ⊢ Ⅎ𝑦𝐷 & ⊢ Ⅎ𝑥𝐶 & ⊢ Ⅎ𝑥𝐷 ⇒ ⊢ (((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝐸 = 𝐹) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹)) | ||
Theorem | iuneq12f 38149 | Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) | ||
Theorem | iineq12f 38150 | Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) | ||
Theorem | opabbi 38151 | Equality deduction for class abstraction of ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓}) | ||
Theorem | mptbi12f 38152 | Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → (𝑥 ∈ 𝐴 ↦ 𝐷) = (𝑥 ∈ 𝐵 ↦ 𝐸)) | ||
Work in progress or things that do not belong anywhere else. | ||
Theorem | orcomdd 38153 | Commutativity of logic disjunction, in double deduction form. Should not be moved to main, see PR #3034 in Github. Use orcomd 871 instead. (Contributed by Giovanni Mascellani, 19-Mar-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 ∨ 𝜒))) | ||
Theorem | scottexf 38154* | A version of scottex 9922 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V | ||
Theorem | scott0f 38155* | A version of scott0 9923 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅) | ||
Theorem | scottn0f 38156* | A version of scott0f 38155 with inequalities instead of equalities. (Contributed by Giovanni Mascellani, 19-Aug-2018.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ≠ ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅) | ||
Theorem | ac6s3f 38157* | Generalization of the Axiom of Choice to classes, with bound-variable hypothesis. (Contributed by Giovanni Mascellani, 19-Aug-2018.) |
⊢ Ⅎ𝑦𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 𝜓) | ||
Theorem | ac6s6 38158* | Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 19-Aug-2018.) |
⊢ Ⅎ𝑦𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓) | ||
Theorem | ac6s6f 38159* | Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 20-Aug-2018.) |
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓) | ||
Syntax | cxrn 38160 | Extend the definition of a class to include the range Cartesian product class. |
class (𝐴 ⋉ 𝐵) | ||
Syntax | ccoss 38161 | Extend the definition of a class to include the class of cosets by a class. (Read: the class of cosets by 𝑅.) |
class ≀ 𝑅 | ||
Syntax | ccoels 38162 | Extend the definition of a class to include the class of coelements on a class. (Read: the class of coelements on 𝐴.) |
class ∼ 𝐴 | ||
Syntax | crels 38163 | Extend the definition of a class to include the relation class. |
class Rels | ||
Syntax | cssr 38164 | Extend the definition of a class to include the subset class. |
class S | ||
Syntax | crefs 38165 | Extend the definition of a class to include the reflexivity class. |
class Refs | ||
Syntax | crefrels 38166 | Extend the definition of a class to include the reflexive relations class. |
class RefRels | ||
Syntax | wrefrel 38167 | Extend the definition of a wff to include the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) |
wff RefRel 𝑅 | ||
Syntax | ccnvrefs 38168 | Extend the definition of a class to include the converse reflexivity class. |
class CnvRefs | ||
Syntax | ccnvrefrels 38169 | Extend the definition of a class to include the converse reflexive relations class. |
class CnvRefRels | ||
Syntax | wcnvrefrel 38170 | Extend the definition of a wff to include the converse reflexive relation predicate. (Read: 𝑅 is a converse reflexive relation.) |
wff CnvRefRel 𝑅 | ||
Syntax | csyms 38171 | Extend the definition of a class to include the symmetry class. |
class Syms | ||
Syntax | csymrels 38172 | Extend the definition of a class to include the symmetry relations class. |
class SymRels | ||
Syntax | wsymrel 38173 | Extend the definition of a wff to include the symmetry relation predicate. (Read: 𝑅 is a symmetric relation.) |
wff SymRel 𝑅 | ||
Syntax | ctrs 38174 | Extend the definition of a class to include the transitivity class (but cf. the transitive class defined in df-tr 5265). |
class Trs | ||
Syntax | ctrrels 38175 | Extend the definition of a class to include the transitive relations class. |
class TrRels | ||
Syntax | wtrrel 38176 | Extend the definition of a wff to include the transitive relation predicate. (Read: 𝑅 is a transitive relation.) |
wff TrRel 𝑅 | ||
Syntax | ceqvrels 38177 | Extend the definition of a class to include the equivalence relations class. |
class EqvRels | ||
Syntax | weqvrel 38178 | Extend the definition of a wff to include the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) |
wff EqvRel 𝑅 | ||
Syntax | ccoeleqvrels 38179 | Extend the definition of a class to include the coelement equivalence relations class. |
class CoElEqvRels | ||
Syntax | wcoeleqvrel 38180 | Extend the definition of a wff to include the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) |
wff CoElEqvRel 𝐴 | ||
Syntax | credunds 38181 | Extend the definition of a class to include the redundancy class. |
class Redunds | ||
Syntax | wredund 38182 | Extend the definition of a wff to include the redundancy predicate. (Read: 𝐴 is redundant with respect to 𝐵 in 𝐶.) |
wff 𝐴 Redund 〈𝐵, 𝐶〉 | ||
Syntax | wredundp 38183 | Extend wff definition to include the redundancy operator for propositions. |
wff redund (𝜑, 𝜓, 𝜒) | ||
Syntax | cdmqss 38184 | Extend the definition of a class to include the domain quotients class. |
class DomainQss | ||
Syntax | wdmqs 38185 | Extend the definition of a wff to include the domain quotient predicate. (Read: the domain quotient of 𝑅 is 𝐴.) |
wff 𝑅 DomainQs 𝐴 | ||
Syntax | cers 38186 | Extend the definition of a class to include the equivalence relations on their domain quotients class. |
class Ers | ||
Syntax | werALTV 38187 | Extend the definition of a wff to include the equivalence relation on its domain quotient predicate. (Read: 𝑅 is an equivalence relation on its domain quotient 𝐴.) |
wff 𝑅 ErALTV 𝐴 | ||
Syntax | ccomembers 38188 | Extend the definition of a class to include the comember equivalence relations class. |
class CoMembErs | ||
Syntax | wcomember 38189 | Extend the definition of a wff to include the comember equivalence relation predicate. (Read: the comember equivalence relation on 𝐴, or, the restricted coelement equivalence relation on its domain quotient 𝐴.) |
wff CoMembEr 𝐴 | ||
Syntax | cfunss 38190 | Extend the definition of a class to include the function set class. |
class Funss | ||
Syntax | cfunsALTV 38191 | Extend the definition of a class to include the functions class, i.e., the function relations class. |
class FunsALTV | ||
Syntax | wfunALTV 38192 | Extend the definition of a wff to include the function predicate, i.e., the function relation predicate. (Read: 𝐹 is a function.) |
wff FunALTV 𝐹 | ||
Syntax | cdisjss 38193 | Extend the definition of a class to include the disjoint set class. |
class Disjss | ||
Syntax | cdisjs 38194 | Extend the definition of a class to include the disjoints class, i.e., the disjoint relations class. |
class Disjs | ||
Syntax | wdisjALTV 38195 | Extend the definition of a wff to include the disjoint predicate, i.e., the disjoint relation predicate. (Read: 𝑅 is a disjoint.) |
wff Disj 𝑅 | ||
Syntax | celdisjs 38196 | Extend the definition of a class to include the disjoint elements class, i.e., the disjoint element relations class. |
class ElDisjs | ||
Syntax | weldisj 38197 | Extend the definition of a wff to include the disjoint element predicate, i.e., the disjoint element relation predicate. (Read: the elements of 𝐴 are disjoint.) |
wff ElDisj 𝐴 | ||
Syntax | wantisymrel 38198 | Extend the definition of a wff to include the antisymmetry relation predicate. (Read: 𝑅 is an antisymmetric relation.) |
wff AntisymRel 𝑅 | ||
Syntax | cparts 38199 | Extend the definition of a class to include the partitions class, i.e., the partition relations class. |
class Parts | ||
Syntax | wpart 38200 | Extend the definition of a wff to include the partition predicate, i.e., the partition relation predicate. (Read: 𝐴 is a partition by 𝑅.) |
wff 𝑅 Part 𝐴 |
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