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Type | Label | Description |
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Statement | ||
Theorem | 19.37vv 38901* | Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑)) | ||
Theorem | 19.28vv 38902* | Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥∀𝑦(𝜓 ∧ 𝜑) ↔ (𝜓 ∧ ∀𝑥∀𝑦𝜑)) | ||
Theorem | pm11.52 38903 | Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓)) | ||
Theorem | 2exanali 38904 | Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (¬ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | aaanv 38905* | Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2206. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓)) | ||
Theorem | pm11.57 38906* | Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) | ||
Theorem | pm11.58 38907* | Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) | ||
Theorem | pm11.59 38908* | Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓))) | ||
Theorem | pm11.6 38909* | Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓)) | ||
Theorem | pm11.61 38910* | Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓)) | ||
Theorem | pm11.62 38911* | Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓 → 𝜒))) | ||
Theorem | pm11.63 38912 | Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (¬ ∃𝑥∃𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | pm11.7 38913 | Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜑) ↔ ∃𝑥∃𝑦𝜑) | ||
Theorem | pm11.71 38914* | Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃)) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))) | ||
Theorem | sbeqal1 38915* | If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧 is true. (Contributed by Andrew Salmon, 2-Jun-2011.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧) | ||
Theorem | sbeqal1i 38916* | Suppose you know 𝑥 = 𝑦 implies 𝑥 = 𝑧, assuming 𝑥 and 𝑧 are distinct. Then, 𝑦 = 𝑧. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧) ⇒ ⊢ 𝑦 = 𝑧 | ||
Theorem | sbeqal2i 38917* | If 𝑥 = 𝑦 implies 𝑥 = 𝑧, then we can infer 𝑧 = 𝑦. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧) ⇒ ⊢ 𝑧 = 𝑦 | ||
Theorem | sbeqalbi 38918* | When both 𝑥 and 𝑧 and 𝑦 and 𝑧 are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.) |
⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦)) | ||
Theorem | axc5c4c711 38919 | Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1762 as the inference rule. This proof extends the idea of axc5c711 34522 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.) |
⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓)) | ||
Theorem | axc5c4c711toc5 38920 | Rederivation of sp 2091 from axc5c4c711 38919. Note that ax6 2287 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1946 instead of ax6 2287, so that this rederivation requires only ax6v 1946 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | axc5c4c711toc4 38921 | Rederivation of axc4 2168 from axc5c4c711 38919. Note that only propositional calculus is required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | axc5c4c711toc7 38922 | Rederivation of axc7 2170 from axc5c4c711 38919. Note that neither axc7 2170 nor ax-11 2074 are required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | ||
Theorem | axc5c4c711to11 38923 | Rederivation of ax-11 2074 from axc5c4c711 38919. Note that ax-11 2074 is not required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | axc11next 38924* | This theorem shows that, given axext4 2635, we can derive a version of axc11n 2342. However, it is weaker than axc11n 2342 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥) | ||
Theorem | pm13.13a 38925 | One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑) | ||
Theorem | pm13.13b 38926 | Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (([𝐴 / 𝑥]𝜑 ∧ 𝑥 = 𝐴) → 𝜑) | ||
Theorem | pm13.14 38927 | Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴) | ||
Theorem | pm13.192 38928* | Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑) | ||
Theorem | pm13.193 38929 | Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦)) | ||
Theorem | pm13.194 38930 | Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦)) | ||
Theorem | pm13.195 38931* | Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3493. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑) | ||
Theorem | pm13.196a 38932* | Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ (¬ 𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 ≠ 𝑥)) | ||
Theorem | 2sbc6g 38933* | Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)) | ||
Theorem | 2sbc5g 38934* | Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)) | ||
Theorem | iotain 38935 | Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑)) | ||
Theorem | iotaexeu 38936 | The iota class exists. This theorem does not require ax-nul 4822 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) | ||
Theorem | iotasbc 38937* | Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define ℩ in terms of a function of (℩𝑥𝜑). Their definition differs in that a function of (℩𝑥𝜑) evaluates to "false" when there isn't a single 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓))) | ||
Theorem | iotasbc2 38938* | Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓) → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) | ||
Theorem | pm14.12 38939* | Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | pm14.122a 38940* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑))) | ||
Theorem | pm14.122b 38941* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑))) | ||
Theorem | pm14.122c 38942* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑))) | ||
Theorem | pm14.123a 38943* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))) | ||
Theorem | pm14.123b 38944* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑))) | ||
Theorem | pm14.123c 38945* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤(𝜑 ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ↔ (∀𝑧∀𝑤(𝜑 → (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)) ∧ ∃𝑧∃𝑤𝜑))) | ||
Theorem | pm14.18 38946 | Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∀𝑥𝜓 → [(℩𝑥𝜑) / 𝑥]𝜓)) | ||
Theorem | iotaequ 38947* | Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (℩𝑥𝑥 = 𝑦) = 𝑦 | ||
Theorem | iotavalb 38948* | Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5900. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦)) | ||
Theorem | iotasbc5 38949* | Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))) | ||
Theorem | pm14.24 38950* | Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = (℩𝑥𝜑))) | ||
Theorem | iotavalsb 38951* | Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓)) | ||
Theorem | sbiota1 38952 | Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓)) | ||
Theorem | sbaniota 38953 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓)) | ||
Theorem | eubi 38954 | Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)) | ||
Theorem | iotasbcq 38955 | Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([(℩𝑥𝜑) / 𝑦]𝜒 ↔ [(℩𝑥𝜓) / 𝑦]𝜒)) | ||
Theorem | elnev 38956* | Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.) |
⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V) | ||
Theorem | rusbcALT 38957 | A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Theorem | compel 38958 | Equivalence between two ways of saying "is a member of the complement of 𝐴." (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) | ||
Theorem | compeq 38959* | Equality between two ways of saying "the complement of 𝐴." (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴} | ||
Theorem | compne 38960 | The complement of 𝐴 is not equal to 𝐴. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.) |
⊢ (V ∖ 𝐴) ≠ 𝐴 | ||
Theorem | compneOLD 38961 | Obsolete proof of compne 38960 as of 11-Nov-2021. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (V ∖ 𝐴) ≠ 𝐴 | ||
Theorem | compab 38962 | Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} | ||
Theorem | conss34OLD 38963 | Obsolete proof of complss 3784 as of 7-Aug-2021. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) | ||
Theorem | conss2 38964 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)) | ||
Theorem | conss1 38965 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
⊢ ((V ∖ 𝐴) ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ 𝐴) | ||
Theorem | ralbidar 38966 | More general form of ralbida 3011. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | rexbidar 38967 | More general form of rexbida 3076. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | dropab1 38968 | Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑥, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ 𝜑}) | ||
Theorem | dropab2 38969 | Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑧, 𝑥〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜑}) | ||
Theorem | ipo0 38970 | If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ ( I Po 𝐴 ↔ 𝐴 = ∅) | ||
Theorem | ifr0 38971 | A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅) | ||
Theorem | ordpss 38972 | ordelpss 5789 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.) |
⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵)) | ||
Theorem | fvsb 38973* | Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (∃!𝑦 𝐴𝐹𝑦 → ([(𝐹‘𝐴) / 𝑥]𝜑 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) | ||
Theorem | fveqsb 38974* | Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.) |
⊢ (𝑥 = (𝐹‘𝐴) → (𝜑 ↔ 𝜓)) & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝜓 ↔ ∃𝑥(∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥) ∧ 𝜑))) | ||
Theorem | xpexb 38975 | A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.) |
⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V) | ||
Theorem | trelpss 38976 | An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5127, ax-reg 8538 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.) |
⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴) | ||
Theorem | addcomgi 38977 | Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) | ||
Syntax | cplusr 38978 | Introduce the operation of vector addition. |
class +𝑟 | ||
Syntax | cminusr 38979 | Introduce the operation of vector subtraction. |
class -𝑟 | ||
Syntax | ctimesr 38980 | Introduce the operation of scalar multiplication. |
class .𝑣 | ||
Syntax | cptdfc 38981 | PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems. |
class PtDf(𝐴, 𝐵) | ||
Syntax | crr3c 38982 | RR3 is a class. |
class RR3 | ||
Syntax | cline3 38983 | line3 is a class. |
class line3 | ||
Definition | df-addr 38984* | Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣)))) | ||
Definition | df-subr 38985* | Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣)))) | ||
Definition | df-mulv 38986* | Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ .𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣)))) | ||
Theorem | addrval 38987* | Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) | ||
Theorem | subrval 38988* | Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) − (𝐵‘𝑣)))) | ||
Theorem | mulvval 38989* | Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) = (𝑣 ∈ ℝ ↦ (𝐴 · (𝐵‘𝑣)))) | ||
Theorem | addrfv 38990 | Vector addition at a value. The operation takes each vector 𝐴 and 𝐵 and forms a new vector whose values are the sum of each of the values of 𝐴 and 𝐵. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶))) | ||
Theorem | subrfv 38991 | Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴-𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) − (𝐵‘𝐶))) | ||
Theorem | mulvfv 38992 | Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴.𝑣𝐵)‘𝐶) = (𝐴 · (𝐵‘𝐶))) | ||
Theorem | addrfn 38993 | Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) Fn ℝ) | ||
Theorem | subrfn 38994 | Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴-𝑟𝐵) Fn ℝ) | ||
Theorem | mulvfn 38995 | Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴.𝑣𝐵) Fn ℝ) | ||
Theorem | addrcom 38996 | Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝐵+𝑟𝐴)) | ||
Definition | df-ptdf 38997* | Define the predicate PtDf, which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.) |
⊢ PtDf(𝐴, 𝐵) = (𝑥 ∈ ℝ ↦ (((𝑥.𝑣(𝐵-𝑟𝐴)) +𝑣 𝐴) “ {1, 2, 3})) | ||
Definition | df-rr3 38998 | Define the set of all points RR3. We define each point 𝐴 as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.) |
⊢ RR3 = (ℝ ↑𝑚 {1, 2, 3}) | ||
Definition | df-line3 38999* | Define the set of all lines. A line is an infinite subset of RR3 that satisfies a PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.) |
⊢ line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2𝑜 ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))} | ||
We are sad to report the passing of long-time contributor Alan Sare (Nov. 9, 1954 - Mar. 23, 2019). Alan's first contribution to Metamath was a shorter proof for tfrlem8 7525 in 2008. He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: http://us.metamath.org/other.html#completeusersproof. His virtual deduction method is explained in the comment for wvd1 39102. Below are some excerpts from his first emails to NM in 2007: ...I have been interested in proving set theory theorems for many years for mental exercise. I enjoy it. I have used a book by Martin Zuckerman. It is informal. I am interested in completely and perfectly proving theorems. Mr. Zuckerman leaves out most of the steps of a proof, of course, like most authors do, as you have noted. A complete proof for higher theorems would require a volume of writing similar to the metamath documents. So I am frustrated when I am not capable of constructing a proof and Zuckerman leaves out steps I do not understand. I could search for the steps in other texts, but I don't do that too much. Metamath may be the answer for me.... ...If we go beyond mathematics, I believe that it is possible to write down all human knowledge in a way similar to the way you have explicated large areas of mathematics. Of course, that would be a much, much more difficult job. For example, it is possible to take a hard science like physics, construct axioms based on experimental results, and to cast all of physics into a collection of axioms and theorems. Maybe this has already been attempted, although I am not familiar with it. When one then moves on to the soft sciences such as social science, this job gets much more difficult. The key is: All human thought consists of logical operations on abstract objects. Usually, these logical operations are done informally. There is no reason why one cannot take any subject and explicate it and take it down to the indivisible postulates in a formal rigorous way.... ...When I read a math book or an engineering book I come across something I don't understand and I am compelled to understand it. But, often it is hopeless. I don't have the time. Or, I would have to read the same thing by multiple authors in the hope that different authors would give parts of the working proof that others have omitted. It is very inefficient. Because I have always been inclined to "get to the bottom" for a 100% fully understood proof.... | ||
Theorem | idiALT 39000 | Placeholder for idi 2. Though unnecessary, this theorem is sometimes used in proofs in this mathbox for pedagogical purposes. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 ⇒ ⊢ 𝜑 |
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