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Type | Label | Description |
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Statement | ||
Theorem | frege81d 42801 | 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 42998. (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 42802 | 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 43000. (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 42803 | 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 43013. (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 42804 | 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 43004. (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 42805 | 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 43008. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege97d 42806 | 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 43014. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈)) ⇒ ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege98d 42807 | 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 43015. (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 42808 | 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 43019. (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 42809 | 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 43023. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege108d 42810 | 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 43025. (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 42811 | 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 43026. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) ⇒ ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege114d 42812 | 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 43031. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)) | ||
Theorem | frege111d 42813 | 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 43028. (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 42814 | 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 43039. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) & ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋)) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege124d 42815 | 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 43041. (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 42816 | 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 43043. (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 42817 | 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 43046. (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 42818 | 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 43048. (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 42819 | 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 43050. (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 3777 for discussion of an example of a class that is not a set. Numbered propositions from [Frege1879]. ax-frege1 42844, ax-frege2 42845, ax-frege8 42863, ax-frege28 42884, ax-frege31 42888, ax-frege41 42899, frege52 (see ax-frege52a 42911, frege52b 42943, and ax-frege52c 42942 for translations), frege54 (see ax-frege54a 42916, frege54b 42947 and ax-frege54c 42946 for translations) and frege58 (see ax-frege58a 42929, ax-frege58b 42955 and frege58c 42975 for translations) are considered "core" or axioms. However, at least ax-frege8 42863 can be derived from ax-frege1 42844 and ax-frege2 42845, see axfrege8 42861. 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 42911, frege52b 42943, and ax-frege52c 42942. In dffrege69 42986, 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 42827 for a definition in terms of image and subset. In dffrege76 42993, 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 43016, 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 43032, 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 43032 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 42800 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 42820, dfxor5 42821. 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 2175 (or simpl 482 and simpr 484 and anifp 1069 in the propositional case) and statements similar to cbvalivw 2009, ax-gen 1796, alrimiv 1929, and alrimdv 1931. 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 1827, 𝐴 = V eqv 3482; Some are not B, ¬ ∀𝑥𝑥 ∈ 𝐵, ∃𝑥¬ 𝑥 ∈ 𝐵 exnal 1828, 𝐵 ⊊ V pssv 4447, 𝐵 ≠ V nev 42824; There are no C, ∀𝑥¬ 𝑥 ∈ 𝐶, ¬ ∃𝑥𝑥 ∈ 𝐶 alnex 1782, 𝐶 = ∅ eq0 4344; There exist D, ¬ ∀𝑥¬ 𝑥 ∈ 𝐷, ∃𝑥𝑥 ∈ 𝐷 df-ex 1781, ∅ ⊊ 𝐷 0pss 4445, 𝐷 ≠ ∅ n0 4347. 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 3972, 𝐸 = (𝐸 ∩ 𝑃) df-ss 3966, 𝐸 ⊆ 𝑃 dfss2 3969; No F are P, ∀𝑥(𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ∈ 𝑃) alinexa 1844, (𝐹 ∩ 𝑃) = ∅ disj1 4451; Some G are not P, ¬ ∀𝑥(𝑥 ∈ 𝐺 → 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐺 ∧ ¬ 𝑥 ∈ 𝑃) exanali 1861, (𝐺 ∩ 𝑃) ⊊ 𝐺 nssinpss 4257, ¬ 𝐺 ⊆ 𝑃 nss 4047; Some H are P, ¬ ∀𝑥(𝑥 ∈ 𝐻 → ¬ 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐻 ∧ 𝑥 ∈ 𝑃) exnalimn 1845, ∅ ⊊ (𝐻 ∩ 𝑃) 0pssin 42825, (𝐻 ∩ 𝑃) ≠ ∅ ndisj 4368. | ||
Theorem | dfxor4 42820 | 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 42821 | 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 42822 | 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 42823 | 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 42824* | Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.) |
⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴) | ||
Theorem | 0pssin 42825* | 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 42826 | The property of relation 𝑅 being hereditary in class 𝐴. |
wff 𝑅 hereditary 𝐴 | ||
Definition | df-he 42827 | The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.) |
⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | dfhe2 42828 | 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 42829* | 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 42830 | 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 42831 | 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 42832 | 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 42833 | 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 42834 | 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 42835 | 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 42836 | The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) |
⊢ ∅ hereditary 𝐴 | ||
Theorem | 0heALT 42837 | 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 42838 | Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.) |
⊢ 𝐴 hereditary ∅ | ||
Theorem | unhe1 42839 | 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 42840 | Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.) |
⊢ {⟨𝐴, 𝐴⟩} hereditary {𝐵} | ||
Theorem | idhe 42841 | The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.) |
⊢ I hereditary 𝐴 | ||
Theorem | psshepw 42842 | 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 42843 | 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 42844 | 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 42845 | 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 42846 | Simplification of triple conjunction. Compare with simp2 1136. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜓))) | ||
Theorem | rp-simp2 42847 | Simplification of triple conjunction. Identical to simp2 1136. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓) | ||
Theorem | rp-frege3g 42848 |
Add antecedent to ax-frege2 42845. More general statement than frege3 42849.
Like ax-frege2 42845, 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 42845 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))) | ||
Theorem | frege3 42849 | Add antecedent to ax-frege2 42845. Special case of rp-frege3g 42848. 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 42850 | Double-use of ax-frege2 42845. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜓)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) | ||
Theorem | rp-frege24 42851 | Introducing an embedded antecedent. Alternate proof for frege24 42869. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))) | ||
Theorem | rp-frege4g 42852 | Deduction related to distribution. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))) | ||
Theorem | frege4 42853 | Special case of closed form of a2d 29. Special case of rp-frege4g 42852. 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 42854 | 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 42855 | Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜃 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) | ||
Theorem | rp-4frege 42856 | Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → ((𝜓 → 𝜑) → 𝜒)) → (𝜑 → 𝜒)) | ||
Theorem | rp-6frege 42857 | Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ((𝜓 → ((𝜒 → 𝜓) → 𝜃)) → (𝜓 → 𝜃))) | ||
Theorem | rp-8frege 42858 | 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 42859 | 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 42860 | 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 42861 |
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 42862 | 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 42863 | 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 42844, and ax-frege2 42845. (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 42864 | 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 42865 | 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 42866 | Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 42854 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 42867 | 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 42868 | 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 42869 | Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 42851 which was proved without relying on ax-frege8 42863. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))) | ||
Theorem | frege16 42870 | 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 42871 | 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 42872 | 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 42873 | 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 42874 | 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 42875 | 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 42876 | 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 42877 | 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 42878 | 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 42879 | 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 42880 | 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 42881 | 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 42882 | 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 42883 | 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 42884 | 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 42885 | 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 42886 | 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 42887 | Identical to notnotr 130. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) |
⊢ (¬ ¬ 𝜑 → 𝜑) | ||
Axiom | ax-frege31 42888 | 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 42889 | 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))) | ||
Theorem | frege33 42890 | If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 146. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) | ||
Theorem | frege34 42891 | If as a consequence of the occurrence of the circumstance 𝜑, when the obstacle 𝜓 is removed, 𝜒 takes place, then from the circumstance that 𝜒 does not take place while 𝜑 occurs the occurrence of the obstacle 𝜓 can be inferred. Closed form of con1d 145. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → 𝜓))) | ||
Theorem | frege35 42892 | Commuted, closed form of con1d 145. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → (¬ 𝜒 → (𝜑 → 𝜓))) | ||
Theorem | frege36 42893 | The case in which 𝜓 is denied, ¬ 𝜑 is affirmed, and 𝜑 is affirmed does not occur. If 𝜑 occurs, then (at least) one of the two, 𝜑 or 𝜓, takes place (no matter what 𝜓 might be). Identical to pm2.24 124. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | ||
Theorem | frege37 42894 | If 𝜒 is a necessary consequence of the occurrence of 𝜓 or 𝜑, then 𝜒 is a necessary consequence of 𝜑 alone. Similar to a closed form of orcs 872. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜒)) | ||
Theorem | frege38 42895 | Identical to pm2.21 123. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | ||
Theorem | frege39 42896 | Syllogism between pm2.18 128 and pm2.24 124. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜑) → (¬ 𝜑 → 𝜓)) | ||
Theorem | frege40 42897 | Anything implies pm2.18 128. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (¬ 𝜑 → ((¬ 𝜓 → 𝜓) → 𝜓)) | ||
Theorem | axfrege41 42898 | Identical to notnot 142. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ¬ ¬ 𝜑) | ||
Axiom | ax-frege41 42899 | The affirmation of 𝜑 denies the denial of 𝜑. Identical to notnot 142. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (𝜑 → ¬ ¬ 𝜑) | ||
Theorem | frege42 42900 | Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ¬ ¬ (𝜑 → 𝜑) |
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