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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfdisjs 36101 | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | ||
Theorem | dfdisjs2 36102 | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I } | ||
Theorem | dfdisjs3 36103* | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑢∀𝑣∀𝑥((𝑢𝑟𝑥 ∧ 𝑣𝑟𝑥) → 𝑢 = 𝑣)} | ||
Theorem | dfdisjs4 36104* | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑥∃*𝑢 𝑢𝑟𝑥} | ||
Theorem | dfdisjs5 36105* | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)} | ||
Theorem | dfdisjALTV 36106 | Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 36097 why we need disjoint relations instead of converse functions anyway. (Contributed by Peter Mazsa, 27-Jul-2021.) |
⊢ ( Disj 𝑅 ↔ ( FunALTV ◡𝑅 ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV2 36107 | Alternate definition of the disjoint relation predicate, cf. dffunALTV2 36081. (Contributed by Peter Mazsa, 27-Jul-2021.) |
⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV3 36108* | Alternate definition of the disjoint relation predicate, cf. dffunALTV3 36082. (Contributed by Peter Mazsa, 28-Jul-2021.) |
⊢ ( Disj 𝑅 ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV4 36109* | Alternate definition of the disjoint relation predicate, cf. dffunALTV4 36083. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ ( Disj 𝑅 ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV5 36110* | Alternate definition of the disjoint relation predicate, cf. dffunALTV5 36084. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅)) | ||
Theorem | dfeldisj2 36111 | Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) |
⊢ ( ElDisj 𝐴 ↔ ≀ ◡(◡ E ↾ 𝐴) ⊆ I ) | ||
Theorem | dfeldisj3 36112* | Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) |
⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ∀𝑥 ∈ (𝑢 ∩ 𝑣)𝑢 = 𝑣) | ||
Theorem | dfeldisj4 36113* | Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) |
⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) | ||
Theorem | dfeldisj5 36114* | Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) |
⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) | ||
Theorem | eldisjs 36115 | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.) |
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs2 36116 | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs3 36117* | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (𝑅 ∈ Disjs ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs4 36118* | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (𝑅 ∈ Disjs ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs5 36119* | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels ))) | ||
Theorem | eldisjsdisj 36120 | The element of the class of disjoint relations and the disjoint relation predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅)) | ||
Theorem | eleldisjs 36121 | Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs )) | ||
Theorem | eleldisjseldisj 36122 | The element of the disjoint elements class and the disjoint elementhood predicate are the same, that is (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴) when 𝐴 is a set. (Contributed by Peter Mazsa, 23-Jul-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴)) | ||
Theorem | disjrel 36123 | Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.) |
⊢ ( Disj 𝑅 → Rel 𝑅) | ||
Theorem | disjss 36124 | Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) |
⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴)) | ||
Theorem | disjssi 36125 | Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ( Disj 𝐵 → Disj 𝐴) | ||
Theorem | disjssd 36126 | Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴)) | ||
Theorem | disjeq 36127 | Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵)) | ||
Theorem | disjeqi 36128 | Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ( Disj 𝐴 ↔ Disj 𝐵) | ||
Theorem | disjeqd 36129 | Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵)) | ||
Theorem | disjdmqseqeq1 36130 | Lemma for the equality theorem for partition ~? parteq1 . (Contributed by Peter Mazsa, 5-Oct-2021.) |
⊢ (𝑅 = 𝑆 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴))) | ||
Theorem | eldisjss 36131 | Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴)) | ||
Theorem | eldisjssi 36132 | Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ( ElDisj 𝐵 → ElDisj 𝐴) | ||
Theorem | eldisjssd 36133 | Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴)) | ||
Theorem | eldisjeq 36134 | Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵)) | ||
Theorem | eldisjeqi 36135 | Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵) | ||
Theorem | eldisjeqd 36136 | Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵)) | ||
Theorem | disjxrn 36137 | Two ways of saying that a range Cartesian product is disjoint. (Contributed by Peter Mazsa, 17-Jun-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) |
⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I ) | ||
Theorem | disjorimxrn 36138 | Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) |
⊢ (( Disj 𝑅 ∨ Disj 𝑆) → Disj (𝑅 ⋉ 𝑆)) | ||
Theorem | disjimxrn 36139 | Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 15-Dec-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) |
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ 𝑆)) | ||
Theorem | disjimres 36140 | Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ( Disj 𝑅 → Disj (𝑅 ↾ 𝐴)) | ||
Theorem | disjimin 36141 | Disjointness condition for intersection. (Contributed by Peter Mazsa, 11-Jun-2021.) (Revised by Peter Mazsa, 28-Sep-2021.) |
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ 𝑆)) | ||
Theorem | disjiminres 36142 | Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴))) | ||
Theorem | disjimxrnres 36143 | Disjointness condition for range Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ (𝑆 ↾ 𝐴))) | ||
Theorem | disjALTV0 36144 | The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ Disj ∅ | ||
Theorem | disjALTVid 36145 | The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.) |
⊢ Disj I | ||
Theorem | disjALTVidres 36146 | The class of identity relations restricted is disjoint. (Contributed by Peter Mazsa, 28-Jun-2020.) (Revised by Peter Mazsa, 27-Sep-2021.) |
⊢ Disj ( I ↾ 𝐴) | ||
Theorem | disjALTVinidres 36147 | The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ Disj (𝑅 ∩ ( I ↾ 𝐴)) | ||
Theorem | disjALTVxrnidres 36148 | The class of range Cartesian product with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 25-Jun-2020.) (Revised by Peter Mazsa, 27-Sep-2021.) |
⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴)) | ||
Theorem | prtlem60 36149 | Lemma for prter3 36178. (Contributed by Rodolfo Medina, 9-Oct-2010.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜓 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) | ||
Theorem | bicomdd 36150 | 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 | jca2r 36151 | 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 36152 | 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 36153 | Lemma for prter3 36178: 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 36154 | Reverse of ibd 272. (Contributed by Rodolfo Medina, 30-Sep-2010.) |
⊢ (𝜑 → (𝜒 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||
Theorem | prtlem100 36155 | Lemma for prter3 36178. (Contributed by Rodolfo Medina, 19-Oct-2010.) |
⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑)) | ||
Theorem | prtlem5 36156* | Lemma for prter1 36175, prter2 36177, prter3 36178 and prtex 36176. (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 36157 | Lemma for prter2 36177. (Contributed by Rodolfo Medina, 17-Oct-2010.) |
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴})) | ||
Theorem | brabsb2 36158* | A closed form of brabsb 5383. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)) | ||
Theorem | eqbrrdv2 36159* | Other version of eqbrrdiv 5631. (Contributed by Rodolfo Medina, 30-Sep-2010.) |
⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) ⇒ ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) | ||
Theorem | prtlem9 36160* | Lemma for prter3 36178. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) | ||
Theorem | prtlem10 36161* | Lemma for prter3 36178. (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 36162 | Lemma for prter2 36177. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) | ||
Theorem | prtlem12 36163* | Lemma for prtex 36176 and prter3 36178. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ ( ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ ) | ||
Theorem | prtlem13 36164* | Lemma for prter1 36175, prter2 36177, prter3 36178 and prtex 36176. (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 36165* | Lemma for prtex 36176, prter2 36177 and prter3 36178. (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 36166* | Lemma for prter2 36177 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 36167 | Extend the definition of a wff to include the partition predicate. |
wff Prt 𝐴 | ||
Definition | df-prt 36168* | Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | ||
Theorem | erprt 36169 | The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ )) | ||
Theorem | prtlem14 36170* | Lemma for prter1 36175, prter2 36177 and prtex 36176. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦))) | ||
Theorem | prtlem15 36171* | Lemma for prter1 36175 and prtex 36176. (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 36172* | Lemma for prter2 36177. (Contributed by Rodolfo Medina, 15-Oct-2010.) |
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥))) | ||
Theorem | prtlem18 36173* | Lemma for prter2 36177. (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 36174* | Lemma for prter2 36177. (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 36175* | 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 36176* | 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 36177* | 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 36178* | 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) | ||
We are sad to report the passing of Metamath creator and long-time contributor Norm Megill (1950 - 2021). Norm of course was the author of the Metamath proof language, the specification, all of the early tools (and some of the later ones), and the foundational work in logic and set theory for set.mm. His tools, now at https://github.com/metamath/metamath-exe , include a proof verifier, a proof assistant, a proof minimizer, style checking and reformatting, and tools for searching and displaying proofs. One of his key insights was that formal proofs can exist not only to be verified by computers, but also to be read by humans. Both the specification of the proof format (which stores full proofs, as opposed to the proof templates used by most proof assistants) and the generated web display of Metamath proofs, one of its distinctive features, contribute to this double objective. Metamath innovated both by using a very simple substitution rule (and then using that to build more complicated notions like free and bound variables) and also by taking the axiom schemas found in many theories and taking them to the next level - by making all axioms, theorems and proofs operate in terms of schemas. Not content to create Metamath for his own amusement, he also published it for the world and encouraged the development of a community of people who contributed to it and created their own tools. He was an active participant in the Metamath mailing list and other forums until days before his passing. It is often our custom to supply a quote from someone memorialized in a mathbox entry. And it is difficult to select a quote for someone who has written so much about Metamath over the years. But here is one quote from the Metamath web page which illustrates not just his clear thinking about what Metamath can and cannot do but also his desire to encourage students at all levels: Q: Will Metamath help me learn abstract mathematics? A: Yes, but probably not by itself. In order to follow a proof in an advanced math textbook, you may need to know prerequisites that could take years to learn. Some people find this frustrating. In contrast, Metamath uses a single, simple substitution rule that allows you to follow any proof mechanically. You can actually jump in anywhere and be convinced that the symbol string you see in a proof step is a consequence of the symbol strings in the earlier steps that it references, even if you don't understand what the symbols mean. But this is quite different from understanding the meaning of the math that results. Metamath alone probably will not give you an intuitive feel for abstract math, in the same way it can be hard to grasp a large computer program just by reading its source code, even though you may understand each individual instruction. However, the Bibliographic Cross-Reference lets you compare informal proofs in math textbooks and see all the implicit missing details "left to the reader." | ||
These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems axc4 , sp 2180, axc7 2325, axc10 2392, axc11 2441, axc11n 2437, axc15 2433, axc9 2389, axc14 2475, and axc16 2259. | ||
Axiom | ax-c5 36179 |
Axiom of Specialization. A universally quantified wff implies the wff
without the universal quantifier (i.e., an instance, or special case, of
the generalized wff). In other words, if something is true for all
𝑥, then 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 1797. Conditional forms of the converse are given by ax-13 2379, ax-c14 36187, ax-c16 36188, and ax-5 1911. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. In our axiomatization, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution (see stdpc4 2073). An interesting alternate axiomatization uses axc5c711 36214 and ax-c4 36180 in place of ax-c5 36179, ax-4 1811, ax-10 2142, and ax-11 2158. This axiom is obsolete and should no longer be used. It is proved above as theorem sp 2180. (Contributed by NM, 3-Jan-1993.) Use sp 2180 instead. (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Axiom | ax-c4 36180 |
Axiom of Quantified Implication. This axiom moves a universal 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 2329. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.) |
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Axiom | ax-c7 36181 |
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 36214 in place
of ax-c5 36179, ax-c7 36181, and ax-11 2158.
This axiom is obsolete and should no longer be used. It is proved above as theorem axc7 2325. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | ||
Axiom | ax-c10 36182 |
A variant of ax6 2391. 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 2392. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Axiom | ax-c11 36183 |
Axiom ax-c11 36183 was the original version of ax-c11n 36184 ("n" for "new"),
before it was discovered (in May 2008) that the shorter ax-c11n 36184 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 2441. (Contributed by NM, 10-May-1993.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Axiom | ax-c11n 36184 |
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 36183 and was replaced with this shorter ax-c11n 36184 ("n" for "new") in May 2008. The old axiom is proved from this one as theorem axc11 2441. Conversely, this axiom is proved from ax-c11 36183 as theorem axc11nfromc11 36222. This axiom was proved redundant in July 2015. See theorem axc11n 2437. This axiom is obsolete and should no longer be used. It is proved above as theorem axc11n 2437. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Axiom | ax-c15 36185 |
Axiom ax-c15 36185 was the original version of ax-12 2175, before it was
discovered (in Jan. 2007) that the shorter ax-12 2175 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 36185 (from which the ax-12 2175 instance follows by theorem ax12 2434.) The proof is by induction on formula length, using ax12eq 36237 and ax12el 36238 for the basis steps and ax12indn 36239, ax12indi 36240, and ax12inda 36244 for the induction steps. (This paragraph is true provided we use ax-c11 36183 in place of ax-c11n 36184.) This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2433, 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 36186 |
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 2389. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
Axiom | ax-c14 36187 |
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 1911; see theorem axc14 2475. Alternately,
ax-5 1911 becomes unnecessary in principle with this
axiom, but we lose the
more powerful metalogic afforded by ax-5 1911.
We retain ax-c14 36187 here to
provide completeness for systems with the simpler metalogic that results
from omitting ax-5 1911, 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 2475. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.) |
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))) | ||
Axiom | ax-c16 36188* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-5 1911
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 5236), but nonetheless it is technically
necessary as you can see from its uses.
This axiom is redundant if we include ax-5 1911; see theorem axc16 2259. Alternately, ax-5 1911 becomes logically redundant in the presence of this axiom, but without ax-5 1911 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 36188 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1911, 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 2259. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | ||
Theorems ax12fromc15 36201 and ax13fromc9 36202 require some intermediate theorems that are included in this section. | ||
Theorem | axc5 36189 | This theorem repeats sp 2180 under the name axc5 36189, so that the Metamath program "MM> VERIFY MARKUP" command will check that it matches axiom scheme ax-c5 36179. (Contributed by NM, 18-Aug-2017.) (Proof modification is discouraged.) Use sp 2180 instead. (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | ax4fromc4 36190 | Rederivation of axiom ax-4 1811 from ax-c4 36180, ax-c5 36179, ax-gen 1797 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2329 for the derivation of ax-c4 36180 from ax-4 1811. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-4 1811 instead. (New usage is discouraged.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | ax10fromc7 36191 | Rederivation of axiom ax-10 2142 from ax-c7 36181, ax-c4 36180, ax-c5 36179, ax-gen 1797 and propositional calculus. See axc7 2325 for the derivation of ax-c7 36181 from ax-10 2142. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-10 2142 instead. (New usage is discouraged.) |
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | ax6fromc10 36192 | Rederivation of axiom ax-6 1970 from ax-c7 36181, ax-c10 36182, ax-gen 1797 and propositional calculus. See axc10 2392 for the derivation of ax-c10 36182 from ax-6 1970. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) Use ax-6 1970 instead. (New usage is discouraged.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | hba1-o 36193 | 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.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Theorem | axc4i-o 36194 | Inference version of ax-c4 36180. (Contributed by NM, 3-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | equid1 36195 | Proof of equid 2019 from our older axioms. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1911; see the proof of equid 2019. See equid1ALT 36221 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑥 = 𝑥 | ||
Theorem | equcomi1 36196 | Proof of equcomi 2024 from equid1 36195, avoiding use of ax-5 1911 (the only use of ax-5 1911 is via ax7 2023, so using ax-7 2015 instead would remove dependency on ax-5 1911). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | ||
Theorem | aecom-o 36197 | Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2438 using ax-c11 36183. Unlike axc11nfromc11 36222, this version does not require ax-5 1911 (see comment of equcomi1 36196). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | aecoms-o 36198 | A commutation rule for identical variable specifiers. Version of aecoms 2439 using ax-c11 36183. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | hbae-o 36199 | All variables are effectively bound in an identical variable specifier. Version of hbae 2442 using ax-c11 36183. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦) | ||
Theorem | dral1-o 36200 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2450 using ax-c11 36183. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
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