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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | exp5g 34501 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((𝜑 ∧ 𝜓) → (((𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp5k 34502 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ (𝜑 → (((𝜓 ∧ (𝜒 ∧ 𝜃)) ∧ 𝜏) → 𝜂)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp56 34503 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp58 34504 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ (((𝜑 ∧ 𝜓) ∧ ((𝜒 ∧ 𝜃) ∧ 𝜏)) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp510 34505 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((𝜑 ∧ (((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏)) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp511 34506 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((𝜑 ∧ ((𝜓 ∧ (𝜒 ∧ 𝜃)) ∧ 𝜏)) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | exp512 34507 | An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.) |
⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏))) → 𝜂) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) | ||
Theorem | 3com12d 34508 | Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.) |
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) ⇒ ⊢ (𝜑 → (𝜒 ∧ 𝜓 ∧ 𝜃)) | ||
Theorem | imp5p 34509 | A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.) |
⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂))) | ||
Theorem | imp5q 34510 | A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.) |
⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)) | ||
Theorem | ecase13d 34511 | Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → ¬ 𝜃) & ⊢ (𝜑 → (𝜒 ∨ 𝜓 ∨ 𝜃)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | subtr 34512 | Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑌 & ⊢ Ⅎ𝑥𝑍 & ⊢ (𝑥 = 𝐴 → 𝑋 = 𝑌) & ⊢ (𝑥 = 𝐵 → 𝑋 = 𝑍) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → 𝑌 = 𝑍)) | ||
Theorem | subtr2 34513 | Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒))) | ||
Theorem | trer 34514* | A relation intersected with its converse is an equivalence relation if the relation is transitive. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (∀𝑎∀𝑏∀𝑐((𝑎 ≤ 𝑏 ∧ 𝑏 ≤ 𝑐) → 𝑎 ≤ 𝑐) → ( ≤ ∩ ◡ ≤ ) Er dom ( ≤ ∩ ◡ ≤ )) | ||
Theorem | elicc3 34515 | An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))) | ||
Theorem | finminlem 34516* | A useful lemma about finite sets. If a property holds for a finite set, it holds for a minimal set. (Contributed by Jeff Hankins, 4-Dec-2009.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ Fin 𝜑 → ∃𝑥(𝜑 ∧ ∀𝑦((𝑦 ⊆ 𝑥 ∧ 𝜓) → 𝑥 = 𝑦))) | ||
Theorem | gtinf 34517* | Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.) (Revised by AV, 10-Oct-2021.) |
⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∧ (𝐴 ∈ ℝ ∧ inf(𝑆, ℝ, < ) < 𝐴)) → ∃𝑧 ∈ 𝑆 𝑧 < 𝐴) | ||
Theorem | opnrebl 34518* | A set is open in the standard topology of the reals precisely when every point can be enclosed in an open ball. (Contributed by Jeff Hankins, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.) |
⊢ (𝐴 ∈ (topGen‘ran (,)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ℝ+ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦)) ⊆ 𝐴)) | ||
Theorem | opnrebl2 34519* | A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.) |
⊢ (𝐴 ∈ (topGen‘ran (,)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ (𝑧 ≤ 𝑦 ∧ ((𝑥 − 𝑧)(,)(𝑥 + 𝑧)) ⊆ 𝐴))) | ||
Theorem | nn0prpwlem 34520* | Lemma for nn0prpw 34521. Use strong induction to show that every positive integer has unique prime power divisors. (Contributed by Jeff Hankins, 28-Sep-2013.) |
⊢ (𝐴 ∈ ℕ → ∀𝑘 ∈ ℕ (𝑘 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴))) | ||
Theorem | nn0prpw 34521* | Two nonnegative integers are the same if and only if they are divisible by the same prime powers. (Contributed by Jeff Hankins, 29-Sep-2013.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) | ||
Theorem | topbnd 34522 | Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) | ||
Theorem | opnbnd 34523 | A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅)) | ||
Theorem | cldbnd 34524 | A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴)) | ||
Theorem | ntruni 34525* | A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∪ 𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂)) | ||
Theorem | clsun 34526 | A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) | ||
Theorem | clsint2 34527* | The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐)) | ||
Theorem | opnregcld 34528* | A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜))) | ||
Theorem | cldregopn 34529* | A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))) | ||
Theorem | neiin 34530 | Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.) |
⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵))) | ||
Theorem | hmeoclda 34531 | Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) (Revised by Mario Carneiro, 3-Jun-2014.) |
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑆) ∈ (Clsd‘𝐾)) | ||
Theorem | hmeocldb 34532 | Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) |
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑆) ∈ (Clsd‘𝐽)) | ||
Theorem | ivthALT 34533* | An alternate proof of the Intermediate Value Theorem ivth 24627 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ 𝐴 < 𝐵 ∧ ((𝐴[,]𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷–cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹‘𝐴)(,)(𝐹‘𝐵))))) → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹‘𝑥) = 𝑈) | ||
Syntax | cfne 34534 | Extend class definition to include the "finer than" relation. |
class Fne | ||
Definition | df-fne 34535* | Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ Fne = {〈𝑥, 𝑦〉 ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))} | ||
Theorem | fnerel 34536 | Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ Rel Fne | ||
Theorem | isfne 34537* | The predicate "𝐵 is finer than 𝐴". This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) | ||
Theorem | isfne4 34538 | The predicate "𝐵 is finer than 𝐴 " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) | ||
Theorem | isfne4b 34539 | A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐵 ∈ 𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)))) | ||
Theorem | isfne2 34540* | The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))) | ||
Theorem | isfne3 34541* | The predicate "𝐵 is finer than 𝐴". (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))) | ||
Theorem | fnebas 34542 | A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝐴Fne𝐵 → 𝑋 = 𝑌) | ||
Theorem | fnetg 34543 | A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝐴Fne𝐵 → 𝐴 ⊆ (topGen‘𝐵)) | ||
Theorem | fnessex 34544* | If 𝐵 is finer than 𝐴 and 𝑆 is an element of 𝐴, every point in 𝑆 is an element of a subset of 𝑆 which is in 𝐵. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝑆) → ∃𝑥 ∈ 𝐵 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑆)) | ||
Theorem | fneuni 34545* | If 𝐵 is finer than 𝐴, every element of 𝐴 is a union of elements of 𝐵. (Contributed by Jeff Hankins, 11-Oct-2009.) |
⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥)) | ||
Theorem | fneint 34546* | If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.) |
⊢ (𝐴Fne𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}) | ||
Theorem | fness 34547 | A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Fne𝐵) | ||
Theorem | fneref 34548 | Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴Fne𝐴) | ||
Theorem | fnetr 34549 | Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ ((𝐴Fne𝐵 ∧ 𝐵Fne𝐶) → 𝐴Fne𝐶) | ||
Theorem | fneval 34550 | Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ ∼ = (Fne ∩ ◡Fne) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∼ 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵))) | ||
Theorem | fneer 34551 | Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ ∼ = (Fne ∩ ◡Fne) ⇒ ⊢ ∼ Er V | ||
Theorem | topfne 34552 | Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽 ⊆ 𝐾 ↔ 𝐽Fne𝐾)) | ||
Theorem | topfneec 34553 | A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ ∼ = (Fne ∩ ◡Fne) ⇒ ⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽)) | ||
Theorem | topfneec2 34554 | A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.) |
⊢ ∼ = (Fne ∩ ◡Fne) ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾)) | ||
Theorem | fnessref 34555* | A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)))) | ||
Theorem | refssfne 34556* | A cover is a refinement iff it is a subcover of something which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
⊢ 𝑋 = ∪ 𝐴 & ⊢ 𝑌 = ∪ 𝐵 ⇒ ⊢ (𝑋 = 𝑌 → (𝐵Ref𝐴 ↔ ∃𝑐(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)))) | ||
Theorem | neibastop1 34557* | A collection of neighborhood bases determines a topology. Part of Theorem 4.5 of Stephen Willard's General Topology. (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅})) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅) & ⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅} ⇒ ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | ||
Theorem | neibastop2lem 34558* | Lemma for neibastop2 34559. (Contributed by Jeff Hankins, 12-Sep-2009.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅})) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅) & ⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅} & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → 𝑥 ∈ 𝑣) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → ∃𝑡 ∈ (𝐹‘𝑥)∀𝑦 ∈ 𝑡 ((𝐹‘𝑦) ∩ 𝒫 𝑣) ≠ ∅) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝑁 ⊆ 𝑋) & ⊢ (𝜑 → 𝑈 ∈ (𝐹‘𝑃)) & ⊢ (𝜑 → 𝑈 ⊆ 𝑁) & ⊢ 𝐺 = (rec((𝑎 ∈ V ↦ ∪ 𝑧 ∈ 𝑎 ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧)), {𝑈}) ↾ ω) & ⊢ 𝑆 = {𝑦 ∈ 𝑋 ∣ ∃𝑓 ∈ ∪ ran 𝐺((𝐹‘𝑦) ∩ 𝒫 𝑓) ≠ ∅} ⇒ ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑁)) | ||
Theorem | neibastop2 34559* | In the topology generated by a neighborhood base, a set is a neighborhood of a point iff it contains a subset in the base. (Contributed by Jeff Hankins, 9-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅})) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅) & ⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅} & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → 𝑥 ∈ 𝑣) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → ∃𝑡 ∈ (𝐹‘𝑥)∀𝑦 ∈ 𝑡 ((𝐹‘𝑦) ∩ 𝒫 𝑣) ≠ ∅) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ 𝑋 ∧ ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅))) | ||
Theorem | neibastop3 34560* | The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅})) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅) & ⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅} & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → 𝑥 ∈ 𝑣) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → ∃𝑡 ∈ (𝐹‘𝑥)∀𝑦 ∈ 𝑡 ((𝐹‘𝑦) ∩ 𝒫 𝑣) ≠ ∅) ⇒ ⊢ (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) | ||
Theorem | topmtcl 34561 | The meet of a collection of topologies on 𝑋 is again a topology on 𝑋. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋)) | ||
Theorem | topmeet 34562* | Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) = ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗}) | ||
Theorem | topjoin 34563* | Two equivalent formulations of the join of a collection of topologies. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) = ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘}) | ||
Theorem | fnemeet1 34564* | The meet of a collection of equivalence classes of covers with respect to fineness. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝐴) | ||
Theorem | fnemeet2 34565* | The meet of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥))) | ||
Theorem | fnejoin1 34566* | Join of equivalence classes under the fineness relation-part one. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆)) | ||
Theorem | fnejoin2 34567* | Join of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇))) | ||
Theorem | fgmin 34568 | Minimality property of a generated filter: every filter that contains 𝐵 contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.) |
⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵 ⊆ 𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹)) | ||
Theorem | neifg 34569* | The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 23002. (Contributed by Jeff Hankins, 3-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋filGen{𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) = ((nei‘𝐽)‘𝑆)) | ||
Theorem | tailfval 34570* | The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ 𝑋 = dom 𝐷 ⇒ ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) | ||
Theorem | tailval 34571 | The tail of an element in a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ 𝑋 = dom 𝐷 ⇒ ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴})) | ||
Theorem | eltail 34572 | An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ 𝑋 = dom 𝐷 ⇒ ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝐶) → (𝐵 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐵)) | ||
Theorem | tailf 34573 | The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ 𝑋 = dom 𝐷 ⇒ ⊢ (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋) | ||
Theorem | tailini 34574 | A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.) |
⊢ 𝑋 = dom 𝐷 ⇒ ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ ((tail‘𝐷)‘𝐴)) | ||
Theorem | tailfb 34575 | The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
⊢ 𝑋 = dom 𝐷 ⇒ ⊢ ((𝐷 ∈ DirRel ∧ 𝑋 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝑋)) | ||
Theorem | filnetlem1 34576* | Lemma for filnet 34580. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) & ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1st ‘𝐵) ⊆ (1st ‘𝐴))) | ||
Theorem | filnetlem2 34577* | Lemma for filnet 34580. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) & ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} ⇒ ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻)) | ||
Theorem | filnetlem3 34578* | Lemma for filnet 34580. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) & ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} ⇒ ⊢ (𝐻 = ∪ ∪ 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))) | ||
Theorem | filnetlem4 34579* | Lemma for filnet 34580. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) & ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} ⇒ ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)))) | ||
Theorem | filnet 34580* | A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)))) | ||
Theorem | tb-ax1 34581 | The first of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | tb-ax2 34582 | The second of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | tb-ax3 34583 |
The third of three axioms in the Tarski-Bernays axiom system.
This axiom, along with ax-mp 5, tb-ax1 34581, and tb-ax2 34582, can be used to derive any theorem or rule that uses only →. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) | ||
Theorem | tbsyl 34584 | The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | re1ax2lem 34585 | Lemma for re1ax2 34586. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | re1ax2 34586 | ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 34581 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | naim1 34587 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒))) | ||
Theorem | naim2 34588 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑))) | ||
Theorem | naim1i 34589 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 ⊼ 𝜒) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | naim2i 34590 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 ⊼ 𝜓) ⇒ ⊢ (𝜒 ⊼ 𝜑) | ||
Theorem | naim12i 34591 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ (𝜓 ⊼ 𝜃) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | nabi1i 34592 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜓 ⊼ 𝜒) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | nabi2i 34593 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ⊼ 𝜓) ⇒ ⊢ (𝜒 ⊼ 𝜑) | ||
Theorem | nabi12i 34594 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) & ⊢ (𝜓 ⊼ 𝜃) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Syntax | w3nand 34595 | The double nand. |
wff (𝜑 ⊼ 𝜓 ⊼ 𝜒) | ||
Definition | df-3nand 34596 | The double nand. This definition allows us to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒))) | ||
Theorem | df3nandALT1 34597 | The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 ⊼ ((𝜓 ⊼ 𝜒) ⊼ (𝜓 ⊼ 𝜒)))) | ||
Theorem | df3nandALT2 34598 | The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | andnand1 34599 | Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ⊼ (𝜑 ⊼ 𝜓 ⊼ 𝜒))) | ||
Theorem | imnand2 34600 | An → nand relation. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) |
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