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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fvrtrcllb0da 43001 | 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 43002 | 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 43003 | 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 43004 | The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.) |
⊢ (r* ∘ t+) = t* | ||
Theorem | cortrcltrcl 43005 | Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (t* ∘ t+) = t* | ||
Theorem | corclrtrcl 43006 | Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (r* ∘ t*) = t* | ||
Theorem | cotrclrcl 43007 | The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.) |
⊢ (t+ ∘ r*) = t* | ||
Theorem | cortrclrcl 43008 | Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (t* ∘ r*) = t* | ||
Theorem | cotrclrtrcl 43009 | Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (t+ ∘ t*) = t* | ||
Theorem | cortrclrtrcl 43010 | 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 43011 | 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 43205. (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 43012 | 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 43209. (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 43013 | 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 43211. (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 43014 | 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 43224. (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 43015 | 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 43215. (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 43016 | 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 43219. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege97d 43017 | 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 43225. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈)) ⇒ ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege98d 43018 | 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 43226. (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 43019 | 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 43230. (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 43020 | 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 43234. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege108d 43021 | 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 43236. (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 43022 | 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 43237. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) ⇒ ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege114d 43023 | 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 43242. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)) | ||
Theorem | frege111d 43024 | 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 43239. (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 43025 | 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 43250. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) & ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋)) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege124d 43026 | 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 43252. (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 43027 | 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 43254. (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 43028 | 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 43257. (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 43029 | 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 43259. (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 43030 | 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 43261. (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 3769 for discussion of an example of a class that is not a set. Numbered propositions from [Frege1879]. ax-frege1 43055, ax-frege2 43056, ax-frege8 43074, ax-frege28 43095, ax-frege31 43099, ax-frege41 43110, frege52 (see ax-frege52a 43122, frege52b 43154, and ax-frege52c 43153 for translations), frege54 (see ax-frege54a 43127, frege54b 43158 and ax-frege54c 43157 for translations) and frege58 (see ax-frege58a 43140, ax-frege58b 43166 and frege58c 43186 for translations) are considered "core" or axioms. However, at least ax-frege8 43074 can be derived from ax-frege1 43055 and ax-frege2 43056, see axfrege8 43072. 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 43122, frege52b 43154, and ax-frege52c 43153. In dffrege69 43197, 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 43038 for a definition in terms of image and subset. In dffrege76 43204, 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 43227, 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 43243, 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 43243 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 43011 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 845, df-an 396, dfxor4 43031, dfxor5 43032. Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biconditional (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 2168 (or simpl 482 and simpr 484 and anifp 1068 in the propositional case) and statements similar to cbvalivw 2002, ax-gen 1789, alrimiv 1922, and alrimdv 1924. 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 1820, 𝐴 = V eqv 3475; Some are not B, ¬ ∀𝑥𝑥 ∈ 𝐵, ∃𝑥¬ 𝑥 ∈ 𝐵 exnal 1821, 𝐵 ⊊ V pssv 4439, 𝐵 ≠ V nev 43035; There are no C, ∀𝑥¬ 𝑥 ∈ 𝐶, ¬ ∃𝑥𝑥 ∈ 𝐶 alnex 1775, 𝐶 = ∅ eq0 4336; There exist D, ¬ ∀𝑥¬ 𝑥 ∈ 𝐷, ∃𝑥𝑥 ∈ 𝐷 df-ex 1774, ∅ ⊊ 𝐷 0pss 4437, 𝐷 ≠ ∅ n0 4339. 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 3964, 𝐸 = (𝐸 ∩ 𝑃) df-ss 3958, 𝐸 ⊆ 𝑃 dfss2 3961; No F are P, ∀𝑥(𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ∈ 𝑃) alinexa 1837, (𝐹 ∩ 𝑃) = ∅ disj1 4443; Some G are not P, ¬ ∀𝑥(𝑥 ∈ 𝐺 → 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐺 ∧ ¬ 𝑥 ∈ 𝑃) exanali 1854, (𝐺 ∩ 𝑃) ⊊ 𝐺 nssinpss 4249, ¬ 𝐺 ⊆ 𝑃 nss 4039; Some H are P, ¬ ∀𝑥(𝑥 ∈ 𝐻 → ¬ 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐻 ∧ 𝑥 ∈ 𝑃) exnalimn 1838, ∅ ⊊ (𝐻 ∩ 𝑃) 0pssin 43036, (𝐻 ∩ 𝑃) ≠ ∅ ndisj 4360. | ||
Theorem | dfxor4 43031 | 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 43032 | 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 43033 | 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 43034 | 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 43035* | Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.) |
⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴) | ||
Theorem | 0pssin 43036* | 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. | ||
Syntax | whe 43037 | The property of relation 𝑅 being hereditary in class 𝐴. |
wff 𝑅 hereditary 𝐴 | ||
Definition | df-he 43038 | The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.) |
⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | dfhe2 43039 | 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 43040* | 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 43041 | 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 43042 | 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 43043 | 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 43044 | 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 43045 | 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 43046 | 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 43047 | The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) |
⊢ ∅ hereditary 𝐴 | ||
Theorem | 0heALT 43048 | 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 43049 | Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.) |
⊢ 𝐴 hereditary ∅ | ||
Theorem | unhe1 43050 | 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 43051 | Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.) |
⊢ {〈𝐴, 𝐴〉} hereditary {𝐵} | ||
Theorem | idhe 43052 | The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.) |
⊢ I hereditary 𝐴 | ||
Theorem | psshepw 43053 | 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 43054 | 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 43055 | 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 43056 | 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 43057 | Simplification of triple conjunction. Compare with simp2 1134. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜓))) | ||
Theorem | rp-simp2 43058 | Simplification of triple conjunction. Identical to simp2 1134. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓) | ||
Theorem | rp-frege3g 43059 |
Add antecedent to ax-frege2 43056. More general statement than frege3 43060.
Like ax-frege2 43056, 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 43056 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))) | ||
Theorem | frege3 43060 | Add antecedent to ax-frege2 43056. Special case of rp-frege3g 43059. 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 43061 | Double-use of ax-frege2 43056. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜓)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) | ||
Theorem | rp-frege24 43062 | Introducing an embedded antecedent. Alternate proof for frege24 43080. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))) | ||
Theorem | rp-frege4g 43063 | Deduction related to distribution. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))) | ||
Theorem | frege4 43064 | Special case of closed form of a2d 29. Special case of rp-frege4g 43063. 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 43065 | 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 43066 | Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜃 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) | ||
Theorem | rp-4frege 43067 | Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → ((𝜓 → 𝜑) → 𝜒)) → (𝜑 → 𝜒)) | ||
Theorem | rp-6frege 43068 | Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ((𝜓 → ((𝜒 → 𝜓) → 𝜃)) → (𝜓 → 𝜃))) | ||
Theorem | rp-8frege 43069 | 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 43070 | 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 43071 | 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 43072 |
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 43073 | 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 43074 | 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 43055, and ax-frege2 43056. (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 43075 | 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 43076 | 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 43077 | Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 43065 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 43078 | 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 43079 | Elimination of a nested antecedent as a partial converse of ja 186. 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 43080 | Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 43062 which was proved without relying on ax-frege8 43074. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))) | ||
Theorem | frege16 43081 | 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 43082 | 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 43083 | 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 43084 | 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 43085 | 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 43086 | 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 43087 | 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 43088 | 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 43089 | 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 43090 | 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 43091 | 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 43092 | 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 43093 | 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 43094 | Contraposition. Identical to con3 153. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | ||
Axiom | ax-frege28 43095 | Contraposition. Identical to con3 153. 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 43096 | Closed form of con3d 152. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓))) | ||
Theorem | frege30 43097 | Commuted, closed form of con3d 152. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑))) | ||
Theorem | axfrege31 43098 | Identical to notnotr 130. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) |
⊢ (¬ ¬ 𝜑 → 𝜑) | ||
Axiom | ax-frege31 43099 | 𝜑 cannot be denied and (at the same time ) ¬ ¬ 𝜑 affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnotr 130. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (¬ ¬ 𝜑 → 𝜑) | ||
Theorem | frege32 43100 | Deduce con1 146 from con3 153. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))) |
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