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Theorem List for Metamath Proof Explorer - 32301-32400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0prpwlem 32301* Lemma for nn0prpw 32302. 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 ¬ ((𝑝𝑛) ∥ 𝑘 ↔ (𝑝𝑛) ∥ 𝐴)))
 
Theoremnn0prpw 32302* 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 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝𝑛) ∥ 𝐴 ↔ (𝑝𝑛) ∥ 𝐵)))
 
20.9.2  Basic topological facts
 
Theoremtopbnd 32303 Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
 
Theoremopnbnd 32304 A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
 
Theoremcldbnd 32305 A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴))
 
Theoremntruni 32306* 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) → 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
 
Theoremclsun 32307 A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
 
Theoremclsint2 32308* 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) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐))
 
Theoremopnregcld 32309* 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 ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
 
Theoremcldregopn 32310* 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 ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
 
Theoremneiin 32311 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)) → (𝑀𝑁) ∈ ((nei‘𝐽)‘(𝐴𝐵)))
 
Theoremhmeoclda 32312 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)) → (𝐹𝑆) ∈ (Clsd‘𝐾))
 
Theoremhmeocldb 32313 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.)
(((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsd‘𝐾)) → (𝐹𝑆) ∈ (Clsd‘𝐽))
 
20.9.3  Topology of the real numbers
 
TheoremivthALT 32314* An alternate proof of the Intermediate Value Theorem ivth 23217 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) ⊆ 𝐷𝐷 ⊆ ℂ ∧ (𝐹 ∈ (𝐷cn→ℂ) ∧ (𝐹 “ (𝐴[,]𝐵)) ⊆ ℝ ∧ 𝑈 ∈ ((𝐹𝐴)(,)(𝐹𝐵))))) → ∃𝑥 ∈ (𝐴(,)𝐵)(𝐹𝑥) = 𝑈)
 
20.9.4  Refinements
 
Syntaxcfne 32315 Extend class definition to include the "finer than" relation.
class Fne
 
Definitiondf-fne 32316* Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.)
Fne = {⟨𝑥, 𝑦⟩ ∣ ( 𝑥 = 𝑦 ∧ ∀𝑧𝑥 𝑧 (𝑦 ∩ 𝒫 𝑧))}
 
Theoremfnerel 32317 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
Rel Fne
 
Theoremisfne 32318* 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       (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴 𝑥 (𝐵 ∩ 𝒫 𝑥))))
 
Theoremisfne4 32319 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 = 𝑌𝐴 ⊆ (topGen‘𝐵)))
 
Theoremisfne4b 32320 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       (𝐵𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵))))
 
Theoremisfne2 32321* 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       (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))))
 
Theoremisfne3 32322* 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       (𝐵𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝐴𝑦(𝑦𝐵𝑥 = 𝑦))))
 
Theoremfnebas 32323 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐴Fne𝐵𝑋 = 𝑌)
 
Theoremfnetg 32324 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝐴Fne𝐵𝐴 ⊆ (topGen‘𝐵))
 
Theoremfnessex 32325* 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.)
((𝐴Fne𝐵𝑆𝐴𝑃𝑆) → ∃𝑥𝐵 (𝑃𝑥𝑥𝑆))
 
Theoremfneuni 32326* If 𝐵 is finer than 𝐴, every element of 𝐴 is a union of elements of 𝐵. (Contributed by Jeff Hankins, 11-Oct-2009.)
((𝐴Fne𝐵𝑆𝐴) → ∃𝑥(𝑥𝐵𝑆 = 𝑥))
 
Theoremfneint 32327* If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
(𝐴Fne𝐵 {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥})
 
Theoremfness 32328 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
𝑋 = 𝐴    &   𝑌 = 𝐵       ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Fne𝐵)
 
Theoremfneref 32329 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
(𝐴𝑉𝐴Fne𝐴)
 
Theoremfnetr 32330 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
((𝐴Fne𝐵𝐵Fne𝐶) → 𝐴Fne𝐶)
 
Theoremfneval 32331 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)       ((𝐴𝑉𝐵𝑊) → (𝐴 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵)))
 
Theoremfneer 32332 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
 
Theoremtopfne 32333 Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
𝑋 = 𝐽    &   𝑌 = 𝐾       ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽𝐾𝐽Fne𝐾))
 
Theoremtopfneec 32334 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))
 
Theoremtopfneec2 32335 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))
 
Theoremfnessref 32336* 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(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
 
Theoremrefssfne 32337* 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(𝐵𝑐 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
 
20.9.5  Neighborhood bases determine topologies
 
Theoremneibastop1 32338* 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.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)    &   𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}       (𝜑𝐽 ∈ (TopOn‘𝑋))
 
Theoremneibastop2lem 32339* Lemma for neibastop2 32340. (Contributed by Jeff Hankins, 12-Sep-2009.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)    &   𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → 𝑥𝑣)    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → ∃𝑡 ∈ (𝐹𝑥)∀𝑦𝑡 ((𝐹𝑦) ∩ 𝒫 𝑣) ≠ ∅)    &   (𝜑𝑃𝑋)    &   (𝜑𝑁𝑋)    &   (𝜑𝑈 ∈ (𝐹𝑃))    &   (𝜑𝑈𝑁)    &   𝐺 = (rec((𝑎 ∈ V ↦ 𝑧𝑎 𝑥𝑋 ((𝐹𝑥) ∩ 𝒫 𝑧)), {𝑈}) ↾ ω)    &   𝑆 = {𝑦𝑋 ∣ ∃𝑓 ran 𝐺((𝐹𝑦) ∩ 𝒫 𝑓) ≠ ∅}       (𝜑 → ∃𝑢𝐽 (𝑃𝑢𝑢𝑁))
 
Theoremneibastop2 32340* 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.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)    &   𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → 𝑥𝑣)    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → ∃𝑡 ∈ (𝐹𝑥)∀𝑦𝑡 ((𝐹𝑦) ∩ 𝒫 𝑣) ≠ ∅)       ((𝜑𝑃𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁𝑋 ∧ ((𝐹𝑃) ∩ 𝒫 𝑁) ≠ ∅)))
 
Theoremneibastop3 32341* The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)    &   𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → 𝑥𝑣)    &   ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → ∃𝑡 ∈ (𝐹𝑥)∀𝑦𝑡 ((𝐹𝑦) ∩ 𝒫 𝑣) ≠ ∅)       (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})
 
20.9.6  Lattice structure of topologies
 
Theoremtopmtcl 32342 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.)
((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) ∈ (TopOn‘𝑋))
 
Theoremtopmeet 32343* 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.)
((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 𝑆) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑘𝑗})
 
Theoremtopjoin 32344* 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.)
((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
 
Theoremfnemeet1 32345* 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.)
((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → (𝒫 𝑋 𝑡𝑆 (topGen‘𝑡))Fne𝐴)
 
Theoremfnemeet2 32346* 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.)
((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (𝑇Fne(𝒫 𝑋 𝑡𝑆 (topGen‘𝑡)) ↔ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑇Fne𝑥)))
 
Theoremfnejoin1 32347* 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.)
((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦𝐴𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, 𝑆))
 
Theoremfnejoin2 32348* 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.)
((𝑋𝑉 ∧ ∀𝑦𝑆 𝑋 = 𝑦) → (if(𝑆 = ∅, {𝑋}, 𝑆)Fne𝑇 ↔ (𝑋 = 𝑇 ∧ ∀𝑥𝑆 𝑥Fne𝑇)))
 
20.9.7  Filter bases
 
Theoremfgmin 32349 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)) → (𝐵𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹))
 
Theoremneifg 32350* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 21640. (Contributed by Jeff Hankins, 3-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → (𝑋filGen{𝑥𝐽𝑆𝑥}) = ((nei‘𝐽)‘𝑆))
 
20.9.8  Directed sets, nets
 
Theoremtailfval 32351* 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) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
 
Theoremtailval 32352 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 ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))
 
Theoremeltail 32353 An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷       ((𝐷 ∈ DirRel ∧ 𝐴𝑋𝐵𝐶) → (𝐵 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐵))
 
Theoremtailf 32354 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)
 
Theoremtailini 32355 A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)
𝑋 = dom 𝐷       ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → 𝐴 ∈ ((tail‘𝐷)‘𝐴))
 
Theoremtailfb 32356 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))
 
Theoremfilnetlem1 32357* Lemma for filnet 32361. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)    &   𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐷𝐵 ↔ ((𝐴𝐻𝐵𝐻) ∧ (1st𝐵) ⊆ (1st𝐴)))
 
Theoremfilnetlem2 32358* Lemma for filnet 32361. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)    &   𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}       (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
 
Theoremfilnetlem3 32359* Lemma for filnet 32361. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)    &   𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}       (𝐻 = 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
 
Theoremfilnetlem4 32360* Lemma for filnet 32361. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)    &   𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}       (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))
 
Theoremfilnet 32361* 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 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))
 
20.10  Mathbox for Anthony Hart
 
20.10.1  Propositional Calculus
 
Theoremtb-ax1 32362 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.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremtb-ax2 32363 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 → (𝜓𝜑))
 
Theoremtb-ax3 32364 The third of three axioms in the Tarski-Bernays axiom system.

This axiom, along with ax-mp 5, tb-ax1 32362, and tb-ax2 32363, 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.)

(((𝜑𝜓) → 𝜑) → 𝜑)
 
Theoremtbsyl 32365 The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theoremre1ax2lem 32366 Lemma for re1ax2 32367. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
 
Theoremre1ax2 32367 ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 32362 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 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremnaim1 32368 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremnaim2 32369 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜒𝜓) → (𝜒𝜑)))
 
Theoremnaim1i 32370 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theoremnaim2i 32371 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜒𝜑)
 
Theoremnaim12i 32372 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓𝜃)       (𝜑𝜒)
 
Theoremnabi1 32373 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theoremnabi2 32374 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
 
Theoremnabi1i 32375 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theoremnabi2i 32376 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜒𝜑)
 
Theoremnabi12i 32377 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓𝜃)       (𝜑𝜒)
 
Syntaxw3nand 32378 The double nand.
wff (𝜑𝜓𝜒)
 
Definitiondf-3nand 32379 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.)
((𝜑𝜓𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
 
Theoremdf3nandALT1 32380 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((𝜑𝜓𝜒) ↔ (𝜑 ⊼ ((𝜓𝜒) ⊼ (𝜓𝜒))))
 
Theoremdf3nandALT2 32381 The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
((𝜑𝜓𝜒) ↔ ¬ (𝜑𝜓𝜒))
 
Theoremandnand1 32382 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓𝜒) ⊼ (𝜑𝜓𝜒)))
 
Theoremimnand2 32383 An nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
((¬ 𝜑𝜓) ↔ ((𝜑𝜑) ⊼ (𝜓𝜓)))
 
20.10.2  Predicate Calculus
 
Theoremallt 32384 For all sets, is true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥
 
Theoremalnof 32385 For all sets, is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥 ¬ ⊥
 
Theoremnalf 32386 Not all sets hold as true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∀𝑥
 
Theoremextt 32387 There exists a set that holds as true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥
 
Theoremnextnt 32388 There does not exist a set, such that is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃𝑥 ¬ ⊤
 
Theoremnextf 32389 There does not exist a set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃𝑥
 
Theoremunnf 32390 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃!𝑥
 
Theoremunnt 32391 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃!𝑥
 
Theoremmont 32392 There does not exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃*𝑥
 
Theoremmof 32393 There exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
∃*𝑥
 
20.10.3  Misc. Single Axiom Systems
 
Theoremmeran1 32394 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))
 
Theoremmeran2 32395 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜏𝜃) ∨ (𝜒 ∨ (𝜑𝜃))))
 
Theoremmeran3 32396 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜒𝜑) ∨ (𝜏 ∨ (𝜃𝜑))))
 
Theoremwaj-ax 32397 A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.)
((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓))))
 
Theoremlukshef-ax2 32398 A single axiom for propositional calculus offered by Lukasiewicz. (Contributed by Anthony Hart, 14-Aug-2011.)
((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜒𝜑)) ⊼ ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 
Theoremarg-ax 32399 ? (Contributed by Anthony Hart, 14-Aug-2011.)
((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜃𝜒) ⊼ ((𝜒𝜃) ⊼ (𝜑𝜃)))))
 
20.10.4  Connective Symmetry
 
Theoremnegsym1 32400 In the paper "On Variable Functors of Propositional Arguments", Lukasiewicz introduced a system that can handle variable connectives. This was done by introducing a variable, marked with a lowercase delta, which takes a wff as input. In the system, "delta 𝜑 " means that "something is true of 𝜑." "delta 𝜑 " can be substituted with ¬ 𝜑, 𝜓𝜑, 𝑥𝜑, etc.

Later on, Meredith discovered a single axiom, in the form of ( delta delta ⊥ → delta 𝜑 ). This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus.

A symmetry with ¬. (Contributed by Anthony Hart, 4-Sep-2011.)

(¬ ¬ ⊥ → ¬ 𝜑)
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