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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sbccom2f 33101* | Commutative law for double class substitution, with non free 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 33102* | Commutative law for double class substitution, with non free 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 | sbcgfi 33103 | Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.) |
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | csbcom2fi 33104* | Commutative law for double class substitution in a class, with non free 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 | ||
Theorem | csbgfi 33105 | Substitution for a variable not free in a class does not affect it, in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.) |
⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵 | ||
A collection of Tseitin axioms used to convert a wff to Conjunctive Normal Form. | ||
Theorem | fald 33106 | Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ¬ ⊥) | ||
Theorem | tsim1 33107 | A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓))) | ||
Theorem | tsim2 33108 | A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜑 ∨ (𝜑 → 𝜓))) | ||
Theorem | tsim3 33109 | A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 → 𝜓))) | ||
Theorem | tsbi1 33110 | A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓))) | ||
Theorem | tsbi2 33111 | A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓))) | ||
Theorem | tsbi3 33112 | A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓))) | ||
Theorem | tsbi4 33113 | A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓))) | ||
Theorem | tsxo1 33114 | A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓))) | ||
Theorem | tsxo2 33115 | A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓))) | ||
Theorem | tsxo3 33116 | A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ⊻ 𝜓))) | ||
Theorem | tsxo4 33117 | A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ (𝜑 ⊻ 𝜓))) | ||
Theorem | tsan1 33118 | A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓))) | ||
Theorem | tsan2 33119 | A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜑 ∨ ¬ (𝜑 ∧ 𝜓))) | ||
Theorem | tsan3 33120 | A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓))) | ||
Theorem | tsna1 33121 | A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓))) | ||
Theorem | tsna2 33122 | A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜑 ∨ (𝜑 ⊼ 𝜓))) | ||
Theorem | tsna3 33123 | A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
⊢ (𝜃 → (𝜓 ∨ (𝜑 ⊼ 𝜓))) | ||
Theorem | tsor1 33124 | A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓))) | ||
Theorem | tsor2 33125 | A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → (¬ 𝜑 ∨ (𝜑 ∨ 𝜓))) | ||
Theorem | tsor3 33126 | A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 ∨ 𝜓))) | ||
Theorem | ts3an1 33127 | 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 33128 | 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 33129 | A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.) |
⊢ (𝜃 → (𝜒 ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))) | ||
Theorem | ts3or1 33130 | 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 33131 | 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 33132 | 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 biimplications) with other constructs. | ||
Theorem | iuneq2f 33133 | Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | abeq12 33134 | Equality deduction for class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) | ||
Theorem | rabeq12f 33135 | 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 33136 | 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 | nfbii2 33137 | Equality deduction for not-freeness. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)) | ||
Theorem | sbeqi 33138 | Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
⊢ ((𝑥 = 𝑦 ∧ ∀𝑧(𝜑 ↔ 𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓)) | ||
Theorem | ralbi12f 33139 | 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 33140 | 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 | mpt2bi123f 33141* | 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 33142 | 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 33143 | 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 33144 | 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 33145 | 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 | scottexf 33146* | A version of scottex 8631 with non-free 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 33147* | A version of scott0 8632 with non-free 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 33148* | A version of scott0f 33147 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 33149* | 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 33150* | 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 33151* | 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) | ||
Theorem | prtlem60 33152 | Lemma for prter3 33185. (Contributed by Rodolfo Medina, 9-Oct-2010.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜓 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) | ||
Theorem | bicomdd 33153 | Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜒))) | ||
Theorem | jca2 33154 | Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜓 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) | ||
Theorem | jca2r 33155 | Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜓 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 ∧ 𝜒))) | ||
Theorem | jca3 33156 | Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → 𝜏) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 ∧ 𝜏)))) | ||
Theorem | prtlem70 33157 | Lemma for prter3 33185: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.) |
⊢ ((((𝜓 ∧ 𝜂) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃 ∧ 𝜏)))) ∧ 𝜂)) | ||
Theorem | ibdr 33158 | Reverse of ibd 257. (Contributed by Rodolfo Medina, 30-Sep-2010.) |
⊢ (𝜑 → (𝜒 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||
Theorem | pm5.31r 33159 | Variant of pm5.31 610. (Contributed by Rodolfo Medina, 15-Oct-2010.) |
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜒 ∧ 𝜓))) | ||
Theorem | 2r19.29 33160 | Double the quantifiers of theorem r19.29. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓)) | ||
Theorem | prtlem100 33161 | Lemma for prter3 33185. (Contributed by Rodolfo Medina, 19-Oct-2010.) |
⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑)) | ||
Theorem | prtlem5 33162* | Lemma for prter1 33182, prter2 33184, prter3 33185 and prtex 33183. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥)) | ||
Theorem | prtlem80 33163 | Lemma for prter2 33184. (Contributed by Rodolfo Medina, 17-Oct-2010.) |
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴})) | ||
Theorem | n0el 33164* | Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥) | ||
Theorem | brabsb2 33165* | A closed form of brabsb 4911. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)) | ||
Theorem | eqbrrdv2 33166* | Other version of eqbrrdiv 5141. (Contributed by Rodolfo Medina, 30-Sep-2010.) |
⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) ⇒ ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) | ||
Theorem | prtlem9 33167* | Lemma for prter3 33185. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) | ||
Theorem | prtlem10 33168* | Lemma for prter3 33185. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ )))) | ||
Theorem | prtlem11 33169 | Lemma for prter2 33184. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) | ||
Theorem | prtlem12 33170* | Lemma for prtex 33183 and prter3 33185. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ ( ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ ) | ||
Theorem | prtlem13 33171* | Lemma for prter1 33182, prter2 33184, prter3 33185 and prtex 33183. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) | ||
Theorem | prtlem16 33172* | Lemma for prtex 33183, prter2 33184 and prter3 33185. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ dom ∼ = ∪ 𝐴 | ||
Theorem | prtlem400 33173* | Lemma for prter2 33184 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ ) | ||
Syntax | wprt 33174 | Extend the definition of a wff to include the partition predicate. |
wff Prt 𝐴 | ||
Definition | df-prt 33175* | Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | ||
Theorem | erprt 33176 | The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ )) | ||
Theorem | prtlem14 33177* | Lemma for prter1 33182, prter2 33184 and prtex 33183. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦))) | ||
Theorem | prtlem15 33178* | Lemma for prter1 33182 and prtex 33183. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → ∃𝑧 ∈ 𝐴 (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧))) | ||
Theorem | prtlem17 33179* | Lemma for prter2 33184. (Contributed by Rodolfo Medina, 15-Oct-2010.) |
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥))) | ||
Theorem | prtlem18 33180* | Lemma for prter2 33184. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤))) | ||
Theorem | prtlem19 33181* | Lemma for prter2 33184. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ )) | ||
Theorem | prter1 33182* | Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴) | ||
Theorem | prtex 33183* | The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ 𝐴 ∈ V)) | ||
Theorem | prter2 33184* | The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → (∪ 𝐴 / ∼ ) = (𝐴 ∖ {∅})) | ||
Theorem | prter3 33185* | For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → ∼ = 𝑆) | ||
Note: A label suffixed with "N" (after the "Atoms..." section below), such as lshpnel2N 33290, means that the definition or theorem is not used for the derivation of hlathil 36271. This is a temporary renaming to assist cleaning up the theorems needed by hlathil 36271. | ||
These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems axc4 , sp 2041, axc7 2117, axc10 2240, axc11 2302, axc11n 2295, axc15 2291, axc9 2290, axc14 2360, and axc16 2120. | ||
Axiom | ax-c5 33186 |
Axiom of Specialization. A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all 𝑥, it is true for any
specific 𝑥 (that would typically occur as a free
variable in the wff
substituted for 𝜑). (A free variable is one that does
not occur in
the scope of a quantifier: 𝑥 and 𝑦 are both free in 𝑥 = 𝑦,
but only 𝑥 is free in ∀𝑦𝑥 = 𝑦.) Axiom scheme C5' in [Megill]
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1713. Conditional forms of the converse are given by ax-13 2234, ax-c14 33194, ax-c16 33195, and ax-5 1827. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 2341. An interesting alternate axiomatization uses axc5c711 33221 and ax-c4 33187 in place of ax-c5 33186, ax-4 1728, ax-10 2006, and ax-11 2021. This axiom is obsolete and should no longer be used. It is proved above as theorem sp 2041. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Axiom | ax-c4 33187 |
Axiom of Quantified Implication. This axiom moves a quantifier from
outside to inside an implication, quantifying 𝜓. Notice that 𝑥
must not be a free variable in the antecedent of the quantified
implication, and we express this by binding 𝜑 to "protect" the
axiom
from a 𝜑 containing a free 𝑥. Axiom
scheme C4' in [Megill]
p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of
[Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.
This axiom is obsolete and should no longer be used. It is proved above as theorem axc4 2115. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.) |
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Axiom | ax-c7 33188 |
Axiom of Quantified Negation. This axiom is used to manipulate negated
quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of
the preprint). An alternate axiomatization could use axc5c711 33221 in place
of ax-c5 33186, ax-c7 33188, and ax-11 2021.
This axiom is obsolete and should no longer be used. It is proved above as theorem axc7 2117. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | ||
Axiom | ax-c10 33189 |
A variant of ax6 2239. Axiom scheme C10' in [Megill] p. 448 (p. 16 of the
preprint).
This axiom is obsolete and should no longer be used. It is proved above as theorem axc10 2240. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Axiom | ax-c11 33190 |
Axiom ax-c11 33190 was the original version of ax-c11n 33191 ("n" for "new"),
before it was discovered (in May 2008) that the shorter ax-c11n 33191 could
replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of
the preprint).
This axiom is obsolete and should no longer be used. It is proved above as theorem axc11 2302. (Contributed by NM, 10-May-1993.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Axiom | ax-c11n 33191 |
Axiom of Quantifier Substitution. One of the equality and substitution
axioms of predicate calculus with equality. Appears as Lemma L12 in
[Megill] p. 445 (p. 12 of the preprint).
The original version of this axiom was ax-c11 33190 and was replaced with this shorter ax-c11n 33191 ("n" for "new") in May 2008. The old axiom is proved from this one as theorem axc11 2302. Conversely, this axiom is proved from ax-c11 33190 as theorem axc11nfromc11 33229. This axiom was proved redundant in July 2015. See theorem axc11n 2295. This axiom is obsolete and should no longer be used. It is proved above as theorem axc11n 2295. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Axiom | ax-c15 33192 |
Axiom ax-c15 33192 was the original version of ax-12 2034, before it was
discovered (in Jan. 2007) that the shorter ax-12 2034 could replace it. It
appears as Axiom scheme C15' in [Megill]
p. 448 (p. 16 of the preprint).
It is based on Lemma 16 of [Tarski] p. 70
and Axiom C8 of [Monk2] p. 105,
from which it can be proved by cases. To understand this theorem more
easily, think of "¬ ∀𝑥𝑥 = 𝑦 →..." as informally meaning
"if
𝑥 and 𝑦 are distinct variables
then..." The antecedent becomes
false if the same variable is substituted for 𝑥 and 𝑦,
ensuring
the theorem is sound whenever this is the case. In some later theorems,
we call an antecedent of the form ¬ ∀𝑥𝑥 = 𝑦 a "distinctor."
Interestingly, if the wff expression substituted for 𝜑 contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-c15 33192 (from which the ax-12 2034 instance follows by theorem ax12 2292.) The proof is by induction on formula length, using ax12eq 33244 and ax12el 33245 for the basis steps and ax12indn 33246, ax12indi 33247, and ax12inda 33251 for the induction steps. (This paragraph is true provided we use ax-c11 33190 in place of ax-c11n 33191.) This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2291, which should be used instead. (Contributed by NM, 14-May-1993.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | ||
Axiom | ax-c9 33193 |
Axiom of Quantifier Introduction. One of the equality and substitution
axioms of predicate calculus with equality. Informally, it says that
whenever 𝑧 is distinct from 𝑥 and
𝑦,
and 𝑥 =
𝑦 is true,
then 𝑥 = 𝑦 quantified with 𝑧 is also
true. In other words, 𝑧
is irrelevant to the truth of 𝑥 = 𝑦. Axiom scheme C9' in [Megill]
p. 448 (p. 16 of the preprint). It apparently does not otherwise appear
in the literature but is easily proved from textbook predicate calculus by
cases.
This axiom is obsolete and should no longer be used. It is proved above as theorem axc9 2290. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
Axiom | ax-c14 33194 |
Axiom of Quantifier Introduction. One of the equality and substitution
axioms for a non-logical predicate in our predicate calculus with
equality. Axiom scheme C14' in [Megill]
p. 448 (p. 16 of the preprint).
It is redundant if we include ax-5 1827; see theorem axc14 2360. Alternately,
ax-5 1827 becomes unnecessary in principle with this
axiom, but we lose the
more powerful metalogic afforded by ax-5 1827.
We retain ax-c14 33194 here to
provide completeness for systems with the simpler metalogic that results
from omitting ax-5 1827, which might be easier to study for some
theoretical
purposes.
This axiom is obsolete and should no longer be used. It is proved above as theorem axc14 2360. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.) |
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))) | ||
Axiom | ax-c16 33195* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-5 1827
to be a
metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p.
16 of the preprint). It apparently does not otherwise appear in the
literature but is easily proved from textbook predicate calculus by
cases. It is a somewhat bizarre axiom since the antecedent is always
false in set theory (see dtru 4783), but nonetheless it is technically
necessary as you can see from its uses.
This axiom is redundant if we include ax-5 1827; see theorem axc16 2120. Alternately, ax-5 1827 becomes logically redundant in the presence of this axiom, but without ax-5 1827 we lose the more powerful metalogic that results from being able to express the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). We retain ax-c16 33195 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1827, which might be easier to study for some theoretical purposes. This axiom is obsolete and should no longer be used. It is proved above as theorem axc16 2120. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | ||
Theorems ax12fromc15 33208 and ax13fromc9 33209 require some intermediate theorems that are included in this section. | ||
Theorem | axc5 33196 | This theorem repeats sp 2041 under the name axc5 33196, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-c5 33186. It is preferred that references to this theorem use the name sp 2041. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | ax4fromc4 33197 | Rederivation of axiom ax-4 1728 from ax-c4 33187, ax-c5 33186, ax-gen 1713 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2115 for the derivation of ax-c4 33187 from ax-4 1728. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | ax10fromc7 33198 | Rederivation of axiom ax-10 2006 from ax-c7 33188, ax-c4 33187, ax-c5 33186, ax-gen 1713 and propositional calculus. See axc7 2117 for the derivation of ax-c7 33188 from ax-10 2006. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | ax6fromc10 33199 | Rederivation of axiom ax-6 1875 from ax-c7 33188, ax-c10 33189, ax-gen 1713 and propositional calculus. See axc10 2240 for the derivation of ax-c10 33189 from ax-6 1875. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | hba1-o 33200 | The setvar 𝑥 is not free in ∀𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
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