P. 442 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44101-44200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	frege91d 44101	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 44304. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
Theorem	frege97d 44102	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 44310. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈))    ⇒   ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)
Theorem	frege98d 44103	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 44311. (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 44104	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 44315. (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 44105	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 44319. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))
Theorem	frege108d 44106	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 44321. (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 44107	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 44322. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))    ⇒   ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)
Theorem	frege114d 44108	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 44327. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴))
Theorem	frege111d 44109	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 44324. (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 44110	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 44335. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝐴 = (𝐹‘𝑋))    &   ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋))    ⇒   ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵))
Theorem	frege124d 44111	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 44337. (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 44112	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 44339. (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 44113	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 44342. (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 44114	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 44344. (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 44115	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 44346. (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))
21.36.6  Propositions from _Begriffsschrift_ 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 3740 for discussion of an example of a class that is not a set. Numbered propositions from [Frege1879]. ax-frege1 44140, ax-frege2 44141, ax-frege8 44159, ax-frege28 44180, ax-frege31 44184, ax-frege41 44195, frege52 (see ax-frege52a 44207, frege52b 44239, and ax-frege52c 44238 for translations), frege54 (see ax-frege54a 44212, frege54b 44243 and ax-frege54c 44242 for translations) and frege58 (see ax-frege58a 44225, ax-frege58b 44251 and frege58c 44271 for translations) are considered "core" or axioms. However, at least ax-frege8 44159 can be derived from ax-frege1 44140 and ax-frege2 44141, see axfrege8 44157. 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 44207, frege52b 44239, and ax-frege52c 44238. In dffrege69 44282, 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 44123 for a definition in terms of image and subset. In dffrege76 44289, 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 44312, 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 44328, 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 44328 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 44096 for an example.
21.36.6.1  _Begriffsschrift_ Chapter I 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 849, df-an 396, dfxor4 44116, dfxor5 44117. 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 arguments 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 2191 (or simpl 482 and simpr 484 and anifp 1072 in the propositional case) and statements similar to cbvalivw 2009, ax-gen 1797, 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 1828, 𝐴 = V eqv 3452; Some are not B, ¬ ∀𝑥𝑥 ∈ 𝐵, ∃𝑥¬ 𝑥 ∈ 𝐵 exnal 1829, 𝐵 ⊊ V pssv 4403, 𝐵 ≠ V nev 44120; There are no C, ∀𝑥¬ 𝑥 ∈ 𝐶, ¬ ∃𝑥𝑥 ∈ 𝐶 alnex 1783, 𝐶 = ∅ eq0 4304; There exist D, ¬ ∀𝑥¬ 𝑥 ∈ 𝐷, ∃𝑥𝑥 ∈ 𝐷 df-ex 1782, ∅ ⊊ 𝐷 0pss 4401, 𝐷 ≠ ∅ n0 4307. 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 3925, 𝐸 = (𝐸 ∩ 𝑃) dfss2 3921, 𝐸 ⊆ 𝑃 df-ss 3920; No F are P, ∀𝑥(𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ∈ 𝑃) alinexa 1845, (𝐹 ∩ 𝑃) = ∅ disj1 4406; Some G are not P, ¬ ∀𝑥(𝑥 ∈ 𝐺 → 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐺 ∧ ¬ 𝑥 ∈ 𝑃) exanali 1861, (𝐺 ∩ 𝑃) ⊊ 𝐺 nssinpss 4221, ¬ 𝐺 ⊆ 𝑃 nss 4000; Some H are P, ¬ ∀𝑥(𝑥 ∈ 𝐻 → ¬ 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐻 ∧ 𝑥 ∈ 𝑃) exnalimn 1846, ∅ ⊊ (𝐻 ∩ 𝑃) 0pssin 44121, (𝐻 ∩ 𝑃) ≠ ∅ ndisj 4324.
Theorem	dfxor4 44116	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 44117	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 44118	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 44119	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 44120*	Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	0pssin 44121*	Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
21.36.6.2  _Begriffsschrift_ Notation hints 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 44122	The property of relation 𝑅 being hereditary in class 𝐴.
wff 𝑅 hereditary 𝐴
Definition	df-he 44123	The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
Theorem	dfhe2 44124	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 44125*	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 44126	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 44127	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 44128	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 44129	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 44130	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 44131	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 44132	The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
⊢ ∅ hereditary 𝐴
Theorem	0heALT 44133	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 44134	Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
⊢ 𝐴 hereditary ∅
Theorem	unhe1 44135	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 44136	Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
⊢ {⟨𝐴, 𝐴⟩} hereditary {𝐵}
Theorem	idhe 44137	The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
⊢ I hereditary 𝐴
Theorem	psshepw 44138	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 44139	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
21.36.6.3  _Begriffsschrift_ Chapter II Implication
Axiom	ax-frege1 44140	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 44141	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 44142	Simplification of triple conjunction. Compare with simp2 1138. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜓)))
Theorem	rp-simp2 44143	Simplification of triple conjunction. Identical to simp2 1138. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓)
Theorem	rp-frege3g 44144	Add antecedent to ax-frege2 44141. More general statement than frege3 44145. Like ax-frege2 44141, 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 44141 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃))))
Theorem	frege3 44145	Add antecedent to ax-frege2 44141. Special case of rp-frege3g 44144. 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 44146	Double-use of ax-frege2 44141. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜓)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
Theorem	rp-frege24 44147	Introducing an embedded antecedent. Alternate proof for frege24 44165. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))
Theorem	rp-frege4g 44148	Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃))))
Theorem	frege4 44149	Special case of closed form of a2d 29. Special case of rp-frege4g 44148. 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 44150	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 44151	Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜃 → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
Theorem	rp-4frege 44152	Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → ((𝜓 → 𝜑) → 𝜒)) → (𝜑 → 𝜒))
Theorem	rp-6frege 44153	Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ((𝜓 → ((𝜒 → 𝜓) → 𝜃)) → (𝜓 → 𝜃)))
Theorem	rp-8frege 44154	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 44155	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 44156	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 44157	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 44158	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 44159	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 44140, and ax-frege2 44141. (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 44160	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 44161	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 44162	Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 44150 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 44163	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 44164	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 44165	Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 44147 which was proved without relying on ax-frege8 44159. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))
Theorem	frege16 44166	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 44167	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 44168	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 44169	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 44170	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 44171	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 44172	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 44173	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 44174	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 44175	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 44176	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 44177	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 44178	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)))))
21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation
Theorem	axfrege28 44179	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 44180	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 44181	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 44182	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 44183	Identical to notnotr 130. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
⊢ (¬ ¬ 𝜑 → 𝜑)
Axiom	ax-frege31 44184	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 44185	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 44186	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 44187	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 44188	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 44189	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 44190	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 876. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜒))
Theorem	frege38 44191	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 44192	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 44193	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 44194	Identical to notnot 142. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ¬ ¬ 𝜑)
Axiom	ax-frege41 44195	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 44196	Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ¬ ¬ (𝜑 → 𝜑)
Theorem	frege43 44197	If there is a choice only between 𝜑 and 𝜑, then 𝜑 takes place. Identical to pm2.18 128. Proposition 43 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	frege44 44198	Similar to a commuted pm2.62 900. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜑))
Theorem	frege45 44199	Deduce pm2.6 191 from con1 146. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))
Theorem	frege46 44200	If 𝜓 holds when 𝜑 occurs as well as when 𝜑 does not occur, then 𝜓 holds. If 𝜓 or 𝜑 occurs and if the occurrences of 𝜑 has 𝜓 as a necessary consequence, then 𝜓 takes place. Identical to pm2.6 191. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))