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Type | Label | Description |
---|---|---|
Statement | ||
Axiom | ax-frege58a 40101 | If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2064. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.) |
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | frege58acor 40102 | Lemma for frege59a 40103. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | frege59a 40103 |
A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition
59 of [Frege1879] p. 51.
Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 40039 incorrectly referenced where frege30 40058 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)))) | ||
Theorem | frege60a 40104 | Swap antecedents of ax-frege58a 40101. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) | ||
Theorem | frege61a 40105 | Lemma for frege65a 40109. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ ((if-(𝜑, 𝜓, 𝜒) → 𝜃) → ((𝜓 ∧ 𝜒) → 𝜃)) | ||
Theorem | frege62a 40106 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (if-(𝜑, 𝜓, 𝜃) → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | frege63a 40107 | Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (if-(𝜑, 𝜓, 𝜃) → (𝜂 → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏)))) | ||
Theorem | frege64a 40108 | Lemma for frege65a 40109. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁)))) | ||
Theorem | frege65a 40109 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) | ||
Theorem | frege66a 40110 | Swap antecedents of frege65a 40109. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) | ||
Theorem | frege67a 40111 | Lemma for frege68a 40112. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ ((((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → (𝜓 ∧ 𝜒))) → (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))) | ||
Theorem | frege68a 40112 | Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))) | ||
Theorem | axfrege52c 40113 | Justification for ax-frege52c 40114. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) | ||
Axiom | ax-frege52c 40114 | One side of dfsbcq 3773. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) | ||
Theorem | frege52b 40115 | The case when the content of 𝑥 is identical with the content of 𝑦 and in which a proposition controlled by an element for which we substitute the content of 𝑥 is affirmed and the same proposition, this time where we substitute the content of 𝑦, is denied does not take place. In [𝑥 / 𝑧]𝜑, 𝑥 can also occur in other than the argument (𝑧) places. Hence 𝑥 may still be contained in [𝑦 / 𝑧]𝜑. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) | ||
Theorem | frege53b 40116 | Lemma for frege102 (via frege92 40181). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑)) | ||
Theorem | axfrege54c 40117 | Reflexive equality of classes. Identical to eqid 2821. Justification for ax-frege54c 40118. (Contributed by RP, 24-Dec-2019.) |
⊢ 𝐴 = 𝐴 | ||
Axiom | ax-frege54c 40118 | Reflexive equality of sets (as classes). Part of Axiom 54 of [Frege1879] p. 50. Identical to eqid 2821. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ 𝐴 = 𝐴 | ||
Theorem | frege54b 40119 | Reflexive equality of sets. The content of 𝑥 is identical with the content of 𝑥. Part of Axiom 54 of [Frege1879] p. 50. Slightly specialized version of eqid 2821. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝑥 = 𝑥 | ||
Theorem | frege54cor1b 40120 | Reflexive equality. (Contributed by RP, 24-Dec-2019.) |
⊢ [𝑥 / 𝑦]𝑦 = 𝑥 | ||
Theorem | frege55lem1b 40121* | Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑 → 𝑥 = 𝑧)) | ||
Theorem | frege55lem2b 40122 | Lemma for frege55b 40123. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥) | ||
Theorem | frege55b 40123 |
Lemma for frege57b 40125. Proposition 55 of [Frege1879] p. 50.
Note that eqtr2 2842 incorporates eqcom 2828 which is stronger than this proposition which is identical to equcomi 2015. Is it possible that Frege tricked himself into assuming what he was out to prove? (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | ||
Theorem | frege56b 40124 | Lemma for frege57b 40125. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))) | ||
Theorem | frege57b 40125 | Analogue of frege57aid 40098. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑)) | ||
Theorem | axfrege58b 40126 | If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2064. Justification for ax-frege58b 40127. (Contributed by RP, 28-Mar-2020.) |
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | ||
Axiom | ax-frege58b 40127 | If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2064. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | ||
Theorem | frege58bid 40128 | If ∀𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2172. See ax-frege58b 40127 and frege58c 40147 for versions which more closely track the original. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | frege58bcor 40129 | Lemma for frege59b 40130. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | ||
Theorem | frege59b 40130 |
A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition
59 of [Frege1879] p. 51.
Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 40039 incorrectly referenced where frege30 40058 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 → (¬ [𝑦 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | frege60b 40131 | Swap antecedents of ax-frege58b 40127. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))) | ||
Theorem | frege61b 40132 | Lemma for frege65b 40136. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (([𝑥 / 𝑦]𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓)) | ||
Theorem | frege62b 40133 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝑦 / 𝑥]𝜓)) | ||
Theorem | frege63b 40134 | Lemma for frege91 40180. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑 → 𝜒) → [𝑦 / 𝑥]𝜒))) | ||
Theorem | frege64b 40135 | Lemma for frege65b 40136. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓 → 𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒))) | ||
Theorem | frege65b 40136 |
A kind of Aristotelian inference. This judgement replaces the mode of
inference barbara 2746 when the minor premise has a general context.
Proposition 65 of [Frege1879] p. 53.
In Frege care is taken to point out that the variables in the first clauses are independent of each other and of the final term so another valid translation could be : ⊢ (∀𝑥([𝑥 / 𝑎]𝜑 → [𝑥 / 𝑏]𝜓) → (∀𝑦([𝑦 / 𝑏]𝜓 → [𝑦 / 𝑐]𝜒) → ([𝑧 / 𝑎]𝜑 → [𝑧 / 𝑐]𝜒))). But that is perhaps too pedantic a translation for this exploration. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))) | ||
Theorem | frege66b 40137 | Swap antecedents of frege65b 40136. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓))) | ||
Theorem | frege67b 40138 | Lemma for frege68b 40139. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))) | ||
Theorem | frege68b 40139 | Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)) | ||
Begriffsschrift Chapter II with equivalence of classes (where they are sets). | ||
Theorem | frege53c 40140 | Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵 → [𝐵 / 𝑥]𝜑)) | ||
Theorem | frege54cor1c 40141* | Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.) |
⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ [𝐴 / 𝑥]𝑥 = 𝐴 | ||
Theorem | frege55lem1c 40142* | Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 𝐵)) | ||
Theorem | frege55lem2c 40143* | Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥) | ||
Theorem | frege55c 40144 | Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥) | ||
Theorem | frege56c 40145* | Lemma for frege57c 40146. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))) | ||
Theorem | frege57c 40146* | Swap order of implication in ax-frege52c 40114. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) | ||
Theorem | frege58c 40147 | Principle related to sp 2172. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) | ||
Theorem | frege59c 40148 |
A kind of Aristotelian inference. Proposition 59 of [Frege1879] p. 51.
Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 40039 incorrectly referenced where frege30 40058 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ ([𝐴 / 𝑥]𝜑 → (¬ [𝐴 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | frege60c 40149 | Swap antecedents of frege58c 40147. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))) | ||
Theorem | frege61c 40150 | Lemma for frege65c 40154. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (([𝐴 / 𝑥]𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | frege62c 40151 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥]𝜓)) | ||
Theorem | frege63c 40152 | Analogue of frege63b 40134. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒))) | ||
Theorem | frege64c 40153 | Lemma for frege65c 40154. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))) | ||
Theorem | frege65c 40154 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))) | ||
Theorem | frege66c 40155 | Swap antecedents of frege65c 40154. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝐴 / 𝑥]𝜒 → [𝐴 / 𝑥]𝜓))) | ||
Theorem | frege67c 40156 | Lemma for frege68c 40157. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑))) | ||
Theorem | frege68c 40157 | Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑)) | ||
(𝑅 “ 𝐴) ⊆ 𝐴 means membership in 𝐴 is hereditary in the sequence dictated by relation 𝑅. This 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. While the above notation is modern, it is cumbersome in the case when 𝐴 is complex and to more closely follow Frege, we abbreviate it with new notation 𝑅 hereditary 𝐴. This greatly shortens the statements for frege97 40186 and frege109 40198. dffrege69 40158 through frege75 40164 develop this, but translation to Metamath is pending some decisions. While Frege does not limit discussion to sets, we may have to depart from Frege by limiting 𝑅 or 𝐴 to sets when we quantify over all hereditary relations or all classes where membership is hereditary in a sequence dictated by 𝑅. | ||
Theorem | dffrege69 40158* | If from the proposition that 𝑥 has property 𝐴 it can be inferred generally, whatever 𝑥 may be, that every result of an application of the procedure 𝑅 to 𝑥 has property 𝐴, then we say " Property 𝐴 is hereditary in the 𝑅-sequence. Definition 69 of [Frege1879] p. 55. (Contributed by RP, 28-Mar-2020.) |
⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴) | ||
Theorem | frege70 40159* | Lemma for frege72 40161. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑉 ⇒ ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))) | ||
Theorem | frege71 40160* | Lemma for frege72 40161. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑉 ⇒ ⊢ ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))) | ||
Theorem | frege72 40161 | If property 𝐴 is hereditary in the 𝑅-sequence, if 𝑥 has property 𝐴, and if 𝑦 is a result of an application of the procedure 𝑅 to 𝑥, then 𝑦 has property 𝐴. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 ⇒ ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) | ||
Theorem | frege73 40162 | Lemma for frege87 40176. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 ⇒ ⊢ ((𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) | ||
Theorem | frege74 40163 | If 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then every result of a application of the procedure 𝑅 to 𝑋 has the property 𝐴. Proposition 74 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 ⇒ ⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) | ||
Theorem | frege75 40164* | If from the proposition that 𝑥 has property 𝐴, whatever 𝑥 may be, it can be inferred that every result of an application of the procedure 𝑅 to 𝑥 has property 𝐴, then property 𝐴 is hereditary in the 𝑅-sequence. Proposition 75 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → 𝑅 hereditary 𝐴) | ||
𝑝(t+‘𝑅)𝑐 means 𝑐 follows 𝑝 in the 𝑅-sequence. dffrege76 40165 through frege98 40187 develop this. This will be shown to be the transitive closure of the relation 𝑅. But more work needs to be done on transitive closure of relations before this is ready for Metamath. | ||
Theorem | dffrege76 40165* |
If from the two propositions that every result of an application of
the procedure 𝑅 to 𝐵 has property 𝑓 and
that property
𝑓 is hereditary in the 𝑅-sequence, it can be inferred,
whatever 𝑓 may be, that 𝐸 has property 𝑓, then
we say
𝐸 follows 𝐵 in the 𝑅-sequence. Definition 76 of
[Frege1879] p. 60.
Each of 𝐵, 𝐸 and 𝑅 must be sets. (Contributed by RP, 2-Jul-2020.) |
⊢ 𝐵 ∈ 𝑈 & ⊢ 𝐸 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 ⇒ ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)) ↔ 𝐵(t+‘𝑅)𝐸) | ||
Theorem | frege77 40166* | If 𝑌 follows 𝑋 in the 𝑅-sequence, if property 𝐴 is hereditary in the 𝑅-sequence, and if every result of an application of the procedure 𝑅 to 𝑋 has the property 𝐴, then 𝑌 has property 𝐴. Proposition 77 of [Frege1879] p. 62. (Contributed by RP, 29-Jun-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴))) | ||
Theorem | frege78 40167* | Commuted form of of frege77 40166. Proposition 78 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴))) | ||
Theorem | frege79 40168* | Distributed form of frege78 40167. Proposition 79 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ ((𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴))) | ||
Theorem | frege80 40169* | Add additional condition to both clauses of frege79 40168. Proposition 80 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ ((𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))) → (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))) | ||
Theorem | frege81 40170 | If 𝑋 has a property 𝐴 that is hereditary in the 𝑅 -sequence, and if 𝑌 follows 𝑋 in the 𝑅-sequence, then 𝑌 has property 𝐴. This is a form of induction attributed to Jakob Bernoulli. Proposition 81 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴))) | ||
Theorem | frege82 40171 | Closed-form deduction based on frege81 40170. Proposition 82 of [Frege1879] p. 64. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ ((𝜑 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝜑 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))) | ||
Theorem | frege83 40172 | Apply commuted form of frege81 40170 when the property 𝑅 is hereditary in a disjunction of two properties, only one of which is known to be held by 𝑋. Proposition 83 of [Frege1879] p. 65. Here we introduce the union of classes where Frege has a disjunction of properties which are represented by membership in either of the classes. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑆 & ⊢ 𝑌 ∈ 𝑇 & ⊢ 𝑅 ∈ 𝑈 & ⊢ 𝐵 ∈ 𝑉 & ⊢ 𝐶 ∈ 𝑊 ⇒ ⊢ (𝑅 hereditary (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐵 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ (𝐵 ∪ 𝐶)))) | ||
Theorem | frege84 40173 | Commuted form of frege81 40170. Proposition 84 of [Frege1879] p. 65. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴))) | ||
Theorem | frege85 40174* | Commuted form of frege77 40166. Proposition 85 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (𝑋(t+‘𝑅)𝑌 → (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴))) | ||
Theorem | frege86 40175* | Conclusion about element one past 𝑌 in the 𝑅-sequence. Proposition 86 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (((𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴))) → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴))))) | ||
Theorem | frege87 40176* | If 𝑍 is a result of an application of the procedure 𝑅 to an object 𝑌 that follows 𝑋 in the 𝑅-sequence and if every result of an application of the procedure 𝑅 to 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then 𝑍 has property 𝐴. Proposition 87 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑍 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝑆 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴)))) | ||
Theorem | frege88 40177* | Commuted form of frege87 40176. Proposition 88 of [Frege1879] p. 67. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑍 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝑆 & ⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → 𝑍 ∈ 𝐴)))) | ||
Theorem | frege89 40178* | One direction of dffrege76 40165. Proposition 89 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 ⇒ ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → 𝑌 ∈ 𝑓)) → 𝑋(t+‘𝑅)𝑌) | ||
Theorem | frege90 40179* | Add antecedent to frege89 40178. Proposition 90 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 ⇒ ⊢ ((𝜑 → ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → 𝑌 ∈ 𝑓))) → (𝜑 → 𝑋(t+‘𝑅)𝑌)) | ||
Theorem | frege91 40180 | Every result of an application of a procedure 𝑅 to an object 𝑋 follows that 𝑋 in the 𝑅-sequence. Proposition 91 of [Frege1879] p. 68. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 ⇒ ⊢ (𝑋𝑅𝑌 → 𝑋(t+‘𝑅)𝑌) | ||
Theorem | frege92 40181 | Inference from frege91 40180. Proposition 92 of [Frege1879] p. 69. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 ⇒ ⊢ (𝑋 = 𝑍 → (𝑋𝑅𝑌 → 𝑍(t+‘𝑅)𝑌)) | ||
Theorem | frege93 40182* | Necessary condition for two elements to be related by the transitive closure. Proposition 93 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 ⇒ ⊢ (∀𝑓(∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝑓) → (𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓)) → 𝑋(t+‘𝑅)𝑌) | ||
Theorem | frege94 40183* | Looking one past a pair related by transitive closure of a relation. Proposition 94 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑍 ∈ 𝑉 & ⊢ 𝑅 ∈ 𝑊 ⇒ ⊢ ((𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → ∀𝑓(∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → (𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓)))) → (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → 𝑋(t+‘𝑅)𝑍))) | ||
Theorem | frege95 40184 | Looking one past a pair related by transitive closure of a relation. Proposition 95 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑍 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝐴 ⇒ ⊢ (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → 𝑋(t+‘𝑅)𝑍)) | ||
Theorem | frege96 40185 | Every result of an application of the procedure 𝑅 to an object that follows 𝑋 in the 𝑅-sequence follows 𝑋 in the 𝑅 -sequence. Proposition 96 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑍 ∈ 𝑊 & ⊢ 𝑅 ∈ 𝐴 ⇒ ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌𝑅𝑍 → 𝑋(t+‘𝑅)𝑍)) | ||
Theorem | frege97 40186 |
The property of following 𝑋 in the 𝑅-sequence is hereditary
in the 𝑅-sequence. Proposition 97 of [Frege1879] p. 71.
Here we introduce the image of a singleton under a relation as class which stands for the property of following 𝑋 in the 𝑅 -sequence. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑅 ∈ 𝑊 ⇒ ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋}) | ||
Theorem | frege98 40187 | If 𝑌 follows 𝑋 and 𝑍 follows 𝑌 in the 𝑅-sequence then 𝑍 follows 𝑋 in the 𝑅-sequence because the transitive closure of a relation has the transitive property. Proposition 98 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 6-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝐴 & ⊢ 𝑌 ∈ 𝐵 & ⊢ 𝑍 ∈ 𝐶 & ⊢ 𝑅 ∈ 𝐷 ⇒ ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍)) | ||
𝑝((t+‘𝑅) ∪ I )𝑐 means 𝑐 is a member of the 𝑅 -sequence begining with 𝑝 and 𝑝 is a member of the 𝑅 -sequence ending with 𝑐. dffrege99 40188 through frege114 40203 develop this. This will be shown to be related to the transitive-reflexive closure of relation 𝑅. But more work needs to be done on transitive closure of relations before this is ready for Metamath. | ||
Theorem | dffrege99 40188 | If 𝑍 is identical with 𝑋 or follows 𝑋 in the 𝑅 -sequence, then we say : "𝑍 belongs to the 𝑅-sequence beginning with 𝑋 " or "𝑋 belongs to the 𝑅-sequence ending with 𝑍". Definition 99 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) |
⊢ 𝑍 ∈ 𝑈 ⇒ ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍) | ||
Theorem | frege100 40189 | One direction of dffrege99 40188. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑍 ∈ 𝑈 ⇒ ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋)) | ||
Theorem | frege101 40190 | Lemma for frege102 40191. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑍 ∈ 𝑈 ⇒ ⊢ ((𝑍 = 𝑋 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)))) | ||
Theorem | frege102 40191 | If 𝑍 belongs to the 𝑅-sequence beginning with 𝑋, then every result of an application of the procedure 𝑅 to 𝑍 follows 𝑋 in the 𝑅-sequence. Proposition 102 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝐴 & ⊢ 𝑍 ∈ 𝐵 & ⊢ 𝑉 ∈ 𝐶 & ⊢ 𝑅 ∈ 𝐷 ⇒ ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) | ||
Theorem | frege103 40192 | Proposition 103 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑍 ∈ 𝑉 ⇒ ⊢ ((𝑍 = 𝑋 → 𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍))) | ||
Theorem | frege104 40193 |
Proposition 104 of [Frege1879] p. 73.
Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑍 ∈ 𝑉 ⇒ ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍)) | ||
Theorem | frege105 40194 | Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑍 ∈ 𝑉 ⇒ ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍) | ||
Theorem | frege106 40195 | Whatever follows 𝑋 in the 𝑅-sequence belongs to the 𝑅 -sequence beginning with 𝑋. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑍 ∈ 𝑉 ⇒ ⊢ (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍) | ||
Theorem | frege107 40196 | Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑉 ∈ 𝐴 ⇒ ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉))) | ||
Theorem | frege108 40197 | If 𝑌 belongs to the 𝑅-sequence beginning with 𝑍, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑍. Proposition 108 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑍 ∈ 𝐴 & ⊢ 𝑌 ∈ 𝐵 & ⊢ 𝑉 ∈ 𝐶 & ⊢ 𝑅 ∈ 𝐷 ⇒ ⊢ (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)) | ||
Theorem | frege109 40198 | The property of belonging to the 𝑅-sequence beginning with 𝑋 is hereditary in the 𝑅-sequence. Proposition 109 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝑈 & ⊢ 𝑅 ∈ 𝑉 ⇒ ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋}) | ||
Theorem | frege110 40199* | Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑋 ∈ 𝐴 & ⊢ 𝑌 ∈ 𝐵 & ⊢ 𝑀 ∈ 𝐶 & ⊢ 𝑅 ∈ 𝐷 ⇒ ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) | ||
Theorem | frege111 40200 | If 𝑌 belongs to the 𝑅-sequence beginning with 𝑍, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑍 or precedes 𝑍 in the 𝑅-sequence. Proposition 111 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Revised by RP, 8-Jul-2020.) (Proof modification is discouraged.) |
⊢ 𝑍 ∈ 𝐴 & ⊢ 𝑌 ∈ 𝐵 & ⊢ 𝑉 ∈ 𝐶 & ⊢ 𝑅 ∈ 𝐷 ⇒ ⊢ (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → (¬ 𝑉(t+‘𝑅)𝑍 → 𝑍((t+‘𝑅) ∪ I )𝑉))) |
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