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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fvrtrcllb0da 40101 | A restriction of the identity relation is a subset of the reflexive-transitive closure of a relation. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*‘𝑅)) | ||
Theorem | fvrtrcllb1d 40102 | A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → 𝑅 ⊆ (t*‘𝑅)) | ||
Theorem | dfrtrcl4 40103 | Reflexive-transitive closure of a relation, expressed as the union of the zeroth power and the transitive closure. (Contributed by RP, 5-Jun-2020.) |
⊢ t* = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (t+‘𝑟))) | ||
Theorem | corcltrcl 40104 | The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.) |
⊢ (r* ∘ t+) = t* | ||
Theorem | cortrcltrcl 40105 | Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (t* ∘ t+) = t* | ||
Theorem | corclrtrcl 40106 | Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (r* ∘ t*) = t* | ||
Theorem | cotrclrcl 40107 | The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.) |
⊢ (t+ ∘ r*) = t* | ||
Theorem | cortrclrcl 40108 | Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (t* ∘ r*) = t* | ||
Theorem | cotrclrtrcl 40109 | Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (t+ ∘ t*) = t* | ||
Theorem | cortrclrtrcl 40110 | The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.) |
⊢ (t* ∘ t*) = t* | ||
Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879]. | ||
Theorem | frege77d 40111 | If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 40306. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) & ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) & ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑈) | ||
Theorem | frege81d 40112 | If the image of 𝑈 is a subset 𝑈, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 81 of [Frege1879] p. 63. Compare with frege81 40310. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) & ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑈) | ||
Theorem | frege83d 40113 | If the image of the union of 𝑈 and 𝑉 is a subset of the union of 𝑈 and 𝑉, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of the union of 𝑈 and 𝑉. Similar to Proposition 83 of [Frege1879] p. 65. Compare with frege83 40312. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) & ⊢ (𝜑 → (𝑅 “ (𝑈 ∪ 𝑉)) ⊆ (𝑈 ∪ 𝑉)) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝑈 ∪ 𝑉)) | ||
Theorem | frege96d 40114 | If 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 40325. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege87d 40115 | If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 87 of [Frege1879] p. 66. Compare with frege87 40316. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) & ⊢ (𝜑 → 𝐶𝑅𝐵) & ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) & ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑈) | ||
Theorem | frege91d 40116 | If 𝐵 follows 𝐴 in 𝑅 then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 91 of [Frege1879] p. 68. Comparw with frege91 40320. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege97d 40117 | If 𝐴 contains all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 97 of [Frege1879] p. 71. Compare with frege97 40326. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈)) ⇒ ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege98d 40118 | If 𝐶 follows 𝐴 and 𝐵 follows 𝐶 in the transitive closure of 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 40327. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) & ⊢ (𝜑 → 𝐶(t+‘𝑅)𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege102d 40119 | If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 40331. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege106d 40120 | If 𝐵 follows 𝐴 in 𝑅, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in 𝑅. Similar to Proposition 106 of [Frege1879] p. 73. Compare with frege106 40335. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege108d 40121 | If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 108 of [Frege1879] p. 74. Compare with frege108 40337. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege109d 40122 | If 𝐴 contains all elements of 𝑈 and all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 109 of [Frege1879] p. 74. Compare with frege109 40338. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) ⇒ ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege114d 40123 | If either 𝑅 relates 𝐴 and 𝐵 or 𝐴 and 𝐵 are the same, then either 𝐴 and 𝐵 are the same, 𝑅 relates 𝐴 and 𝐵, 𝑅 relates 𝐵 and 𝐴. Similar to Proposition 114 of [Frege1879] p. 76. Compare with frege114 40343. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)) | ||
Theorem | frege111d 40124 | If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 40340. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴)) | ||
Theorem | frege122d 40125 | If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 is the successor of 𝑋, then 𝐴 and 𝐵 are the same (or 𝐵 follows 𝐴 in the transitive closure of 𝐹). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 40351. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) & ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋)) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege124d 40126 | If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 40353. (Contributed by RP, 16-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) & ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) & ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege126d 40127 | If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 126 of [Frege1879] p. 81. Compare with frege126 40355. (Contributed by RP, 16-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) & ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) & ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) | ||
Theorem | frege129d 40128 | If 𝐹 is a function and (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹, the successor of 𝐴 is either 𝐵 or it follows 𝐵 or it comes before 𝐵 in the transitive closure of 𝐹. Similar to Proposition 129 of [Frege1879] p. 83. Comparw with frege129 40358. (Contributed by RP, 16-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ dom 𝐹) & ⊢ (𝜑 → 𝐶 = (𝐹‘𝐴)) & ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐵(t+‘𝐹)𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶(t+‘𝐹)𝐵)) | ||
Theorem | frege131d 40129 | If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴. Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 40360. (Contributed by RP, 17-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege133d 40130 | If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops). Similar to Proposition 133 of [Frege1879] p. 86. Compare with frege133 40362. (Contributed by RP, 18-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐴) & ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) | ||
In 1879, Frege introduced notation for documenting formal reasoning about propositions (and classes) which covered elements of propositional logic, predicate calculus and reasoning about relations. However, due to the pitfalls of naive set theory, adapting this work for inclusion in set.mm required dividing statements about propositions from those about classes and identifying when a restriction to sets is required. For an overview comparing the details of Frege's two-dimensional notation and that used in set.mm, see mmfrege.html. See ru 3771 for discussion of an example of a class that is not a set. Numbered propositions from [Frege1879]. ax-frege1 40156, ax-frege2 40157, ax-frege8 40175, ax-frege28 40196, ax-frege31 40200, ax-frege41 40211, frege52 (see ax-frege52a 40223, frege52b 40255, and ax-frege52c 40254 for translations), frege54 (see ax-frege54a 40228, frege54b 40259 and ax-frege54c 40258 for translations) and frege58 (see ax-frege58a 40241, ax-frege58b 40267 and frege58c 40287 for translations) are considered "core" or axioms. However, at least ax-frege8 40175 can be derived from ax-frege1 40156 and ax-frege2 40157, see axfrege8 40173. Frege introduced implication, negation and the universal quantifier as primitives and did not in the numbered propositions use other logical connectives other than equivalence introduced in ax-frege52a 40223, frege52b 40255, and ax-frege52c 40254. In dffrege69 40298, Frege introduced 𝑅 hereditary 𝐴 to say that relation 𝑅, when restricted to operate on elements of class 𝐴, will only have elements of class 𝐴 in its domain; see df-he 40139 for a definition in terms of image and subset. In dffrege76 40305, Frege introduced notation for the concept of two sets related by the transitive closure of a relation, for which we write 𝑋(t+‘𝑅)𝑌, which requires 𝑅 to also be a set. In dffrege99 40328, Frege introduced notation for the concept of two sets either identical or related by the transitive closure of a relation, for which we write 𝑋((t+‘𝑅) ∪ I )𝑌, which is a superclass of sets related by the reflexive-transitive relation 𝑋(t*‘𝑅)𝑌. Finally, in dffrege115 40344, Frege introduced notation for the concept of a relation having the property elements in its domain pair up with only one element each in its range, for which we write Fun ◡◡𝑅 (to ignore any non-relational content of the class 𝑅). Frege did this without the expressing concept of a relation (or its transitive closure) as a class, and needed to invent conventions for discussing indeterminate propositions with two slots free and how to recognize which of the slots was domain and which was range. See mmfrege.html 40344 for details. English translations for specific propositions lifted in part from a translation by Stefan Bauer-Mengelberg as reprinted in From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. An attempt to align these propositions in the larger set.mm database has also been made. See frege77d 40111 for an example. | ||
Section 2 introduces the turnstile ⊢ which turns an idea which may be true 𝜑 into an assertion that it does hold true ⊢ 𝜑. Section 5 introduces implication, (𝜑 → 𝜓). Section 6 introduces the single rule of interference relied upon, modus ponens ax-mp 5. Section 7 introduces negation and with in synonyms for or (¬ 𝜑 → 𝜓), and ¬ (𝜑 → ¬ 𝜓), and two for exclusive-or corresponding to df-or 844, df-an 399, dfxor4 40131, dfxor5 40132. Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biimplication (𝜑 ↔ 𝜓) or class equality 𝐴 = 𝐵 in this adaptation. Section 10 introduces "truth functions" for one or two variables in equally troubling notation, as the arguments may be understood to be logical predicates or collections. Here f(𝜑) is interpreted to mean if-(𝜑, 𝜓, 𝜒) where the content of the "function" is specified by the latter two argments or logical equivalent, while g(𝐴) is read as 𝐴 ∈ 𝐺 and h(𝐴, 𝐵) as 𝐴𝐻𝐵. This necessarily introduces a need for set theory as both 𝐴 ∈ 𝐺 and 𝐴𝐻𝐵 cannot hold unless 𝐴 is a set. (Also 𝐵.) Section 11 introduces notation for generality, but there is no standard notation for generality when the variable is a proposition because it was realized after Frege that the universe of all possible propositions includes paradoxical constructions leading to the failure of naive set theory. So adopting f(𝜑) as if-(𝜑, 𝜓, 𝜒) would result in the translation of ∀𝜑 f (𝜑) as (𝜓 ∧ 𝜒). For collections, we must generalize over set variables or run into the same problems; this leads to ∀𝐴 g(𝐴) being translated as ∀𝑎𝑎 ∈ 𝐺 and so forth. Under this interpreation the text of section 11 gives us sp 2182 (or simpl 485 and simpr 487 and anifp 1065 in the propositional case) and statements similar to cbvalivw 2014, ax-gen 1796, alrimiv 1928, and alrimdv 1930. These last four introduce a generality and have no useful definition in terms of propositional variables. Section 12 introduces some combinations of primitive symbols and their human language counterparts. Using class notation, these can also be expressed without dummy variables. All are A, ∀𝑥𝑥 ∈ 𝐴, ¬ ∃𝑥¬ 𝑥 ∈ 𝐴 alex 1826, 𝐴 = V eqv 3502; Some are not B, ¬ ∀𝑥𝑥 ∈ 𝐵, ∃𝑥¬ 𝑥 ∈ 𝐵 exnal 1827, 𝐵 ⊊ V pssv 4398, 𝐵 ≠ V nev 40135; There are no C, ∀𝑥¬ 𝑥 ∈ 𝐶, ¬ ∃𝑥𝑥 ∈ 𝐶 alnex 1782, 𝐶 = ∅ eq0 4308; There exist D, ¬ ∀𝑥¬ 𝑥 ∈ 𝐷, ∃𝑥𝑥 ∈ 𝐷 df-ex 1781, ∅ ⊊ 𝐷 0pss 4396, 𝐷 ≠ ∅ n0 4310. Notation for relations between expressions also can be written in various ways. All E are P, ∀𝑥(𝑥 ∈ 𝐸 → 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐸 ∧ ¬ 𝑥 ∈ 𝑃) dfss6 3957, 𝐸 = (𝐸 ∩ 𝑃) df-ss 3952, 𝐸 ⊆ 𝑃 dfss2 3955; No F are P, ∀𝑥(𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ∈ 𝑃) alinexa 1843, (𝐹 ∩ 𝑃) = ∅ disj1 4401; Some G are not P, ¬ ∀𝑥(𝑥 ∈ 𝐺 → 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐺 ∧ ¬ 𝑥 ∈ 𝑃) exanali 1859, (𝐺 ∩ 𝑃) ⊊ 𝐺 nssinpss 4233, ¬ 𝐺 ⊆ 𝑃 nss 4029; Some H are P, ¬ ∀𝑥(𝑥 ∈ 𝐻 → ¬ 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐻 ∧ 𝑥 ∈ 𝑃) exnalimn 1844, ∅ ⊊ (𝐻 ∩ 𝑃) 0pssin 40136, (𝐻 ∩ 𝑃) ≠ ∅ ndisj 4327. | ||
Theorem | dfxor4 40131 | Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓))) | ||
Theorem | dfxor5 40132 | Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓))) | ||
Theorem | df3or2 40133 | Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 → (¬ 𝜓 → 𝜒))) | ||
Theorem | df3an2 40134 | Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒))) | ||
Theorem | nev 40135* | Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.) |
⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴) | ||
Theorem | 0pssin 40136* | Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.) |
⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | ||
The statement 𝑅 hereditary 𝐴 means relation 𝑅 is hereditary (in the sense of Frege) in the class 𝐴 or (𝑅 “ 𝐴) ⊆ 𝐴. The former is only a slight reduction in the number of symbols, but this reduces the number of floating hypotheses needed to be checked. As Frege was not using the language of classes or sets, this naturally differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on. | ||
Theorem | rp-imass 40137 | If the 𝑅-image of a class 𝐴 is a subclass of 𝐵, then the restriction of 𝑅 to 𝐴 is a subset of the Cartesian product of 𝐴 and 𝐵. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) | ||
Syntax | whe 40138 | The property of relation 𝑅 being hereditary in class 𝐴. |
wff 𝑅 hereditary 𝐴 | ||
Definition | df-he 40139 | The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.) |
⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | dfhe2 40140 | The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.) |
⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐴)) | ||
Theorem | dfhe3 40141* | The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.) |
⊢ (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))) | ||
Theorem | heeq12 40142 | Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑆 hereditary 𝐵)) | ||
Theorem | heeq1 40143 | Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
⊢ (𝑅 = 𝑆 → (𝑅 hereditary 𝐴 ↔ 𝑆 hereditary 𝐴)) | ||
Theorem | heeq2 40144 | Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
⊢ (𝐴 = 𝐵 → (𝑅 hereditary 𝐴 ↔ 𝑅 hereditary 𝐵)) | ||
Theorem | sbcheg 40145 | Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | hess 40146 | Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
⊢ (𝑆 ⊆ 𝑅 → (𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴)) | ||
Theorem | xphe 40147 | Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.) |
⊢ (𝐴 × 𝐵) hereditary 𝐵 | ||
Theorem | 0he 40148 | The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) |
⊢ ∅ hereditary 𝐴 | ||
Theorem | 0heALT 40149 | The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ∅ hereditary 𝐴 | ||
Theorem | he0 40150 | Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.) |
⊢ 𝐴 hereditary ∅ | ||
Theorem | unhe1 40151 | The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.) |
⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → (𝑅 ∪ 𝑆) hereditary 𝐴) | ||
Theorem | snhesn 40152 | Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.) |
⊢ {〈𝐴, 𝐴〉} hereditary {𝐵} | ||
Theorem | idhe 40153 | The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.) |
⊢ I hereditary 𝐴 | ||
Theorem | psshepw 40154 | The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.) |
⊢ ◡ [⊊] hereditary 𝒫 𝐴 | ||
Theorem | sshepw 40155 | The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.) |
⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴 | ||
Axiom | ax-frege1 40156 | The case in which 𝜑 is denied, 𝜓 is affirmed, and 𝜑 is affirmed is excluded. This is evident since 𝜑 cannot at the same time be denied and affirmed. Axiom 1 of [Frege1879] p. 26. Identical to ax-1 6. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Axiom | ax-frege2 40157 | If a proposition 𝜒 is a necessary consequence of two propositions 𝜓 and 𝜑 and one of those, 𝜓, is in turn a necessary consequence of the other, 𝜑, then the proposition 𝜒 is a necessary consequence of the latter one, 𝜑, alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | rp-simp2-frege 40158 | Simplification of triple conjunction. Compare with simp2 1133. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜓))) | ||
Theorem | rp-simp2 40159 | Simplification of triple conjunction. Identical to simp2 1133. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓) | ||
Theorem | rp-frege3g 40160 |
Add antecedent to ax-frege2 40157. More general statement than frege3 40161.
Like ax-frege2 40157, it is essentially a closed form of mpd 15,
however it
has an extra antecedent.
It would be more natural to prove from a1i 11 and ax-frege2 40157 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))) | ||
Theorem | frege3 40161 | Add antecedent to ax-frege2 40157. Special case of rp-frege3g 40160. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜑 → 𝜓)) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))) | ||
Theorem | rp-misc1-frege 40162 | Double-use of ax-frege2 40157. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜓)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) | ||
Theorem | rp-frege24 40163 | Introducing an embedded antecedent. Alternate proof for frege24 40181. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))) | ||
Theorem | rp-frege4g 40164 | Deduction related to distribution. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))) | ||
Theorem | frege4 40165 | Special case of closed form of a2d 29. Special case of rp-frege4g 40164. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))) | ||
Theorem | frege5 40166 | A closed form of syl 17. Identical to imim2 58. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) | ||
Theorem | rp-7frege 40167 | Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜃 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) | ||
Theorem | rp-4frege 40168 | Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → ((𝜓 → 𝜑) → 𝜒)) → (𝜑 → 𝜒)) | ||
Theorem | rp-6frege 40169 | Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ((𝜓 → ((𝜒 → 𝜓) → 𝜃)) → (𝜓 → 𝜃))) | ||
Theorem | rp-8frege 40170 | Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → ((𝜒 → 𝜓) → 𝜃))) → (𝜑 → (𝜓 → 𝜃))) | ||
Theorem | rp-frege25 40171 | Closed form for a1dd 50. Alternate route to Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) | ||
Theorem | frege6 40172 | A closed form of imim2d 57 which is a deduction adding nested antecedents. Proposition 6 of [Frege1879] p. 33. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → ((𝜃 → 𝜓) → (𝜃 → 𝜒)))) | ||
Theorem | axfrege8 40173 |
Swap antecedents. Identical to pm2.04 90. This demonstrates that Axiom 8
of [Frege1879] p. 35 is redundant.
Proof follows closely proof of pm2.04 90 in https://us.metamath.org/mmsolitaire/pmproofs.txt 90, but in the style of Frege's 1879 work. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | frege7 40174 | A closed form of syl6 35. The first antecedent is used to replace the consequent of the second antecedent. Proposition 7 of [Frege1879] p. 34. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜃 → 𝜑)) → (𝜒 → (𝜃 → 𝜓)))) | ||
Axiom | ax-frege8 40175 | Swap antecedents. If two conditions have a proposition as a consequence, their order is immaterial. Third axiom of Frege's 1879 work but identical to pm2.04 90 which can be proved from only ax-mp 5, ax-frege1 40156, and ax-frege2 40157. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | frege26 40176 | Identical to idd 24. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜓)) | ||
Theorem | frege27 40177 | We cannot (at the same time) affirm 𝜑 and deny 𝜑. Identical to id 22. Proposition 27 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝜑) | ||
Theorem | frege9 40178 | Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 40166 only in an unessential way. Identical to imim1 83. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | frege12 40179 | A closed form of com23 86. Proposition 12 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃)))) | ||
Theorem | frege11 40180 | Elimination of a nested antecedent as a partial converse of ja 188. If the proposition that 𝜓 takes place or 𝜑 does not is a sufficient condition for 𝜒, then 𝜓 by itself is a sufficient condition for 𝜒. Identical to jarr 106. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)) | ||
Theorem | frege24 40181 | Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 40163 which was proved without relying on ax-frege8 40175. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))) | ||
Theorem | frege16 40182 | A closed form of com34 91. Proposition 16 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) → (𝜑 → (𝜓 → (𝜃 → (𝜒 → 𝜏))))) | ||
Theorem | frege25 40183 | Closed form for a1dd 50. Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜃 → 𝜒)))) | ||
Theorem | frege18 40184 | Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜓 → (𝜃 → 𝜒)))) | ||
Theorem | frege22 40185 | A closed form of com45 97. Proposition 22 of [Frege1879] p. 41. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) → (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃 → 𝜂)))))) | ||
Theorem | frege10 40186 | Result commuting antecedents within an antecedent. Proposition 10 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜃) → ((𝜓 → (𝜑 → 𝜒)) → 𝜃)) | ||
Theorem | frege17 40187 | A closed form of com3l 89. Proposition 17 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜓 → (𝜒 → (𝜑 → 𝜃)))) | ||
Theorem | frege13 40188 | A closed form of com3r 87. Proposition 13 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃)))) | ||
Theorem | frege14 40189 | Closed form of a deduction based on com3r 87. Proposition 14 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) → (𝜑 → (𝜃 → (𝜓 → (𝜒 → 𝜏))))) | ||
Theorem | frege19 40190 | A closed form of syl6 35. Proposition 19 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜒 → 𝜃) → (𝜑 → (𝜓 → 𝜃)))) | ||
Theorem | frege23 40191 | Syllogism followed by rotation of three antecedents. Proposition 23 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜏 → 𝜑) → (𝜓 → (𝜒 → (𝜏 → 𝜃))))) | ||
Theorem | frege15 40192 | A closed form of com4r 94. Proposition 15 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) → (𝜃 → (𝜑 → (𝜓 → (𝜒 → 𝜏))))) | ||
Theorem | frege21 40193 | Replace antecedent in antecedent. Proposition 21 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜃) → ((𝜃 → 𝜓) → 𝜒))) | ||
Theorem | frege20 40194 | A closed form of syl8 76. Proposition 20 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜃 → 𝜏) → (𝜑 → (𝜓 → (𝜒 → 𝜏))))) | ||
Theorem | axfrege28 40195 | Contraposition. Identical to con3 156. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | ||
Axiom | ax-frege28 40196 | Contraposition. Identical to con3 156. Theorem *2.16 of [WhiteheadRussell] p. 103. Axiom 28 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | ||
Theorem | frege29 40197 | Closed form of con3d 155. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓))) | ||
Theorem | frege30 40198 | Commuted, closed form of con3d 155. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑))) | ||
Theorem | axfrege31 40199 | Identical to notnotr 132. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) |
⊢ (¬ ¬ 𝜑 → 𝜑) | ||
Axiom | ax-frege31 40200 | 𝜑 cannot be denied and (at the same time ) ¬ ¬ 𝜑 affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnotr 132. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (¬ ¬ 𝜑 → 𝜑) |
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