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Theorem List for Metamath Proof Explorer - 35201-35300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopnrebl2 35201* 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 ∈ (topGenβ€˜ran (,)) ↔ (𝐴 βŠ† ℝ ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ (𝑧 ≀ 𝑦 ∧ ((π‘₯ βˆ’ 𝑧)(,)(π‘₯ + 𝑧)) βŠ† 𝐴)))
 
Theoremnn0prpwlem 35202* Lemma for nn0prpw 35203. Use strong induction to show that every positive integer has unique prime power divisors. (Contributed by Jeff Hankins, 28-Sep-2013.)
(𝐴 ∈ β„• β†’ βˆ€π‘˜ ∈ β„• (π‘˜ < 𝐴 β†’ βˆƒπ‘ ∈ β„™ βˆƒπ‘› ∈ β„• Β¬ ((𝑝↑𝑛) βˆ₯ π‘˜ ↔ (𝑝↑𝑛) βˆ₯ 𝐴)))
 
Theoremnn0prpw 35203* 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 ∈ β„•0 ∧ 𝐡 ∈ β„•0) β†’ (𝐴 = 𝐡 ↔ βˆ€π‘ ∈ β„™ βˆ€π‘› ∈ β„• ((𝑝↑𝑛) βˆ₯ 𝐴 ↔ (𝑝↑𝑛) βˆ₯ 𝐡)))
 
21.12.2  Basic topological facts
 
Theoremtopbnd 35204 Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))) = (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄)))
 
Theoremopnbnd 35205 A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (𝐴 ∈ 𝐽 ↔ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)))) = βˆ…))
 
Theoremcldbnd 35206 A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (𝐴 ∈ (Clsdβ€˜π½) ↔ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))) βŠ† 𝐴))
 
Theoremntruni 35207* 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 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝑂 βŠ† 𝒫 𝑋) β†’ βˆͺ π‘œ ∈ 𝑂 ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂))
 
Theoremclsun 35208 A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) = (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))
 
Theoremclsint2 35209* The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐢 βŠ† 𝒫 𝑋) β†’ ((clsβ€˜π½)β€˜βˆ© 𝐢) βŠ† ∩ 𝑐 ∈ 𝐢 ((clsβ€˜π½)β€˜π‘))
 
Theoremopnregcld 35210* A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴 ↔ βˆƒπ‘œ ∈ 𝐽 𝐴 = ((clsβ€˜π½)β€˜π‘œ)))
 
Theoremcldregopn 35211* A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴 ↔ βˆƒπ‘ ∈ (Clsdβ€˜π½)𝐴 = ((intβ€˜π½)β€˜π‘)))
 
Theoremneiin 35212 Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.)
((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ (𝑀 ∩ 𝑁) ∈ ((neiβ€˜π½)β€˜(𝐴 ∩ 𝐡)))
 
Theoremhmeoclda 35213 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
(((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (𝐹 β€œ 𝑆) ∈ (Clsdβ€˜πΎ))
 
Theoremhmeocldb 35214 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.)
(((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsdβ€˜πΎ)) β†’ (◑𝐹 β€œ 𝑆) ∈ (Clsdβ€˜π½))
 
21.12.3  Topology of the real numbers
 
TheoremivthALT 35215* An alternate proof of the Intermediate Value Theorem ivth 24970 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 ∧ 𝐡 ∈ ℝ ∧ π‘ˆ ∈ ℝ) ∧ 𝐴 < 𝐡 ∧ ((𝐴[,]𝐡) βŠ† 𝐷 ∧ 𝐷 βŠ† β„‚ ∧ (𝐹 ∈ (𝐷–cnβ†’β„‚) ∧ (𝐹 β€œ (𝐴[,]𝐡)) βŠ† ℝ ∧ π‘ˆ ∈ ((πΉβ€˜π΄)(,)(πΉβ€˜π΅))))) β†’ βˆƒπ‘₯ ∈ (𝐴(,)𝐡)(πΉβ€˜π‘₯) = π‘ˆ)
 
21.12.4  Refinements
 
Syntaxcfne 35216 Extend class definition to include the "finer than" relation.
class Fne
 
Definitiondf-fne 35217* Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.)
Fne = {⟨π‘₯, π‘¦βŸ© ∣ (βˆͺ π‘₯ = βˆͺ 𝑦 ∧ βˆ€π‘§ ∈ π‘₯ 𝑧 βŠ† βˆͺ (𝑦 ∩ 𝒫 𝑧))}
 
Theoremfnerel 35218 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
Rel Fne
 
Theoremisfne 35219* The predicate "𝐡 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 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐡 ∈ 𝐢 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝐴 π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯))))
 
Theoremisfne4 35220 The predicate "𝐡 is finer than 𝐴 " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ 𝐴 βŠ† (topGenβ€˜π΅)))
 
Theoremisfne4b 35221 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 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐡 ∈ 𝑉 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ (topGenβ€˜π΄) βŠ† (topGenβ€˜π΅))))
 
Theoremisfne2 35222* The predicate "𝐡 is finer than 𝐴". (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐡 ∈ 𝐢 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝐡 (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† π‘₯))))
 
Theoremisfne3 35223* The predicate "𝐡 is finer than 𝐴". (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐡 ∈ 𝐢 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))))
 
Theoremfnebas 35224 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐴Fne𝐡 β†’ 𝑋 = π‘Œ)
 
Theoremfnetg 35225 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝐴Fne𝐡 β†’ 𝐴 βŠ† (topGenβ€˜π΅))
 
Theoremfnessex 35226* If 𝐡 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 𝐡. (Contributed by Jeff Hankins, 28-Sep-2009.)
((𝐴Fne𝐡 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝑆) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑆))
 
Theoremfneuni 35227* If 𝐡 is finer than 𝐴, every element of 𝐴 is a union of elements of 𝐡. (Contributed by Jeff Hankins, 11-Oct-2009.)
((𝐴Fne𝐡 ∧ 𝑆 ∈ 𝐴) β†’ βˆƒπ‘₯(π‘₯ βŠ† 𝐡 ∧ 𝑆 = βˆͺ π‘₯))
 
Theoremfneint 35228* If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
(𝐴Fne𝐡 β†’ ∩ {π‘₯ ∈ 𝐡 ∣ 𝑃 ∈ π‘₯} βŠ† ∩ {π‘₯ ∈ 𝐴 ∣ 𝑃 ∈ π‘₯})
 
Theoremfness 35229 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   ((𝐡 ∈ 𝐢 ∧ 𝐴 βŠ† 𝐡 ∧ 𝑋 = π‘Œ) β†’ 𝐴Fne𝐡)
 
Theoremfneref 35230 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
(𝐴 ∈ 𝑉 β†’ 𝐴Fne𝐴)
 
Theoremfnetr 35231 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
((𝐴Fne𝐡 ∧ 𝐡Fne𝐢) β†’ 𝐴Fne𝐢)
 
Theoremfneval 35232 Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)
∼ = (Fne ∩ β—‘Fne)    β‡’   ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴 ∼ 𝐡 ↔ (topGenβ€˜π΄) = (topGenβ€˜π΅)))
 
Theoremfneer 35233 Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
∼ = (Fne ∩ β—‘Fne)    β‡’    ∼ Er V
 
Theoremtopfne 35234 Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
𝑋 = βˆͺ 𝐽    &   π‘Œ = βˆͺ 𝐾    β‡’   ((𝐾 ∈ Top ∧ 𝑋 = π‘Œ) β†’ (𝐽 βŠ† 𝐾 ↔ 𝐽Fne𝐾))
 
Theoremtopfneec 35235 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 ∩ β—‘Fne)    β‡’   (𝐽 ∈ Top β†’ (𝐴 ∈ [𝐽] ∼ ↔ (topGenβ€˜π΄) = 𝐽))
 
Theoremtopfneec2 35236 A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)
∼ = (Fne ∩ β—‘Fne)    β‡’   ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) β†’ ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾))
 
Theoremfnessref 35237* 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 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝑋 = π‘Œ β†’ (𝐴Fne𝐡 ↔ βˆƒπ‘(𝑐 βŠ† 𝐡 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
 
Theoremrefssfne 35238* 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 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝑋 = π‘Œ β†’ (𝐡Ref𝐴 ↔ βˆƒπ‘(𝐡 βŠ† 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
 
21.12.5  Neighborhood bases determine topologies
 
Theoremneibastop1 35239* 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.)
(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆ(𝒫 𝒫 𝑋 βˆ– {βˆ…}))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯) ∧ 𝑀 ∈ (πΉβ€˜π‘₯))) β†’ ((πΉβ€˜π‘₯) ∩ 𝒫 (𝑣 ∩ 𝑀)) β‰  βˆ…)    &   π½ = {π‘œ ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘œ ((πΉβ€˜π‘₯) ∩ 𝒫 π‘œ) β‰  βˆ…}    β‡’   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
 
Theoremneibastop2lem 35240* Lemma for neibastop2 35241. (Contributed by Jeff Hankins, 12-Sep-2009.)
(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆ(𝒫 𝒫 𝑋 βˆ– {βˆ…}))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯) ∧ 𝑀 ∈ (πΉβ€˜π‘₯))) β†’ ((πΉβ€˜π‘₯) ∩ 𝒫 (𝑣 ∩ 𝑀)) β‰  βˆ…)    &   π½ = {π‘œ ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘œ ((πΉβ€˜π‘₯) ∩ 𝒫 π‘œ) β‰  βˆ…}    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ π‘₯ ∈ 𝑣)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ βˆƒπ‘‘ ∈ (πΉβ€˜π‘₯)βˆ€π‘¦ ∈ 𝑑 ((πΉβ€˜π‘¦) ∩ 𝒫 𝑣) β‰  βˆ…)    &   (πœ‘ β†’ 𝑃 ∈ 𝑋)    &   (πœ‘ β†’ 𝑁 βŠ† 𝑋)    &   (πœ‘ β†’ π‘ˆ ∈ (πΉβ€˜π‘ƒ))    &   (πœ‘ β†’ π‘ˆ βŠ† 𝑁)    &   πΊ = (rec((π‘Ž ∈ V ↦ βˆͺ 𝑧 ∈ π‘Ž βˆͺ π‘₯ ∈ 𝑋 ((πΉβ€˜π‘₯) ∩ 𝒫 𝑧)), {π‘ˆ}) β†Ύ Ο‰)    &   π‘† = {𝑦 ∈ 𝑋 ∣ βˆƒπ‘“ ∈ βˆͺ ran 𝐺((πΉβ€˜π‘¦) ∩ 𝒫 𝑓) β‰  βˆ…}    β‡’   (πœ‘ β†’ βˆƒπ‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑁))
 
Theoremneibastop2 35241* 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.)
(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆ(𝒫 𝒫 𝑋 βˆ– {βˆ…}))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯) ∧ 𝑀 ∈ (πΉβ€˜π‘₯))) β†’ ((πΉβ€˜π‘₯) ∩ 𝒫 (𝑣 ∩ 𝑀)) β‰  βˆ…)    &   π½ = {π‘œ ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘œ ((πΉβ€˜π‘₯) ∩ 𝒫 π‘œ) β‰  βˆ…}    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ π‘₯ ∈ 𝑣)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ βˆƒπ‘‘ ∈ (πΉβ€˜π‘₯)βˆ€π‘¦ ∈ 𝑑 ((πΉβ€˜π‘¦) ∩ 𝒫 𝑣) β‰  βˆ…)    β‡’   ((πœ‘ ∧ 𝑃 ∈ 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) ↔ (𝑁 βŠ† 𝑋 ∧ ((πΉβ€˜π‘ƒ) ∩ 𝒫 𝑁) β‰  βˆ…)))
 
Theoremneibastop3 35242* The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆ(𝒫 𝒫 𝑋 βˆ– {βˆ…}))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯) ∧ 𝑀 ∈ (πΉβ€˜π‘₯))) β†’ ((πΉβ€˜π‘₯) ∩ 𝒫 (𝑣 ∩ 𝑀)) β‰  βˆ…)    &   π½ = {π‘œ ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘œ ((πΉβ€˜π‘₯) ∩ 𝒫 π‘œ) β‰  βˆ…}    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ π‘₯ ∈ 𝑣)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ βˆƒπ‘‘ ∈ (πΉβ€˜π‘₯)βˆ€π‘¦ ∈ 𝑑 ((πΉβ€˜π‘¦) ∩ 𝒫 𝑣) β‰  βˆ…)    β‡’   (πœ‘ β†’ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)βˆ€π‘₯ ∈ 𝑋 ((neiβ€˜π‘—)β€˜{π‘₯}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((πΉβ€˜π‘₯) ∩ 𝒫 𝑛) β‰  βˆ…})
 
21.12.6  Lattice structure of topologies
 
Theoremtopmtcl 35243 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 βŠ† (TopOnβ€˜π‘‹)) β†’ (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOnβ€˜π‘‹))
 
Theoremtopmeet 35244* 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 βŠ† (TopOnβ€˜π‘‹)) β†’ (𝒫 𝑋 ∩ ∩ 𝑆) = βˆͺ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗})
 
Theoremtopjoin 35245* 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 βŠ† (TopOnβ€˜π‘‹)) β†’ (topGenβ€˜(fiβ€˜({𝑋} βˆͺ βˆͺ 𝑆))) = ∩ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 𝑗 βŠ† π‘˜})
 
Theoremfnemeet1 35246* 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 ∧ βˆ€π‘¦ ∈ 𝑆 𝑋 = βˆͺ 𝑦 ∧ 𝐴 ∈ 𝑆) β†’ (𝒫 𝑋 ∩ ∩ 𝑑 ∈ 𝑆 (topGenβ€˜π‘‘))Fne𝐴)
 
Theoremfnemeet2 35247* 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 ∧ βˆ€π‘¦ ∈ 𝑆 𝑋 = βˆͺ 𝑦) β†’ (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑑 ∈ 𝑆 (topGenβ€˜π‘‘)) ↔ (𝑋 = βˆͺ 𝑇 ∧ βˆ€π‘₯ ∈ 𝑆 𝑇Fneπ‘₯)))
 
Theoremfnejoin1 35248* 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 ∧ βˆ€π‘¦ ∈ 𝑆 𝑋 = βˆͺ 𝑦 ∧ 𝐴 ∈ 𝑆) β†’ 𝐴Fneif(𝑆 = βˆ…, {𝑋}, βˆͺ 𝑆))
 
Theoremfnejoin2 35249* 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 ∧ βˆ€π‘¦ ∈ 𝑆 𝑋 = βˆͺ 𝑦) β†’ (if(𝑆 = βˆ…, {𝑋}, βˆͺ 𝑆)Fne𝑇 ↔ (𝑋 = βˆͺ 𝑇 ∧ βˆ€π‘₯ ∈ 𝑆 π‘₯Fne𝑇)))
 
21.12.7  Filter bases
 
Theoremfgmin 35250 Minimality property of a generated filter: every filter that contains 𝐡 contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
((𝐡 ∈ (fBasβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐡 βŠ† 𝐹 ↔ (𝑋filGen𝐡) βŠ† 𝐹))
 
Theoremneifg 35251* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 23345. (Contributed by Jeff Hankins, 3-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑋filGen{π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯}) = ((neiβ€˜π½)β€˜π‘†))
 
21.12.8  Directed sets, nets
 
Theoremtailfval 35252* The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷    β‡’   (𝐷 ∈ DirRel β†’ (tailβ€˜π·) = (π‘₯ ∈ 𝑋 ↦ (𝐷 β€œ {π‘₯})))
 
Theoremtailval 35253 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    β‡’   ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) β†’ ((tailβ€˜π·)β€˜π΄) = (𝐷 β€œ {𝐴}))
 
Theoremeltail 35254 An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷    β‡’   ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝐢) β†’ (𝐡 ∈ ((tailβ€˜π·)β€˜π΄) ↔ 𝐴𝐷𝐡))
 
Theoremtailf 35255 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    β‡’   (𝐷 ∈ DirRel β†’ (tailβ€˜π·):π‘‹βŸΆπ’« 𝑋)
 
Theoremtailini 35256 A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)
𝑋 = dom 𝐷    β‡’   ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ ((tailβ€˜π·)β€˜π΄))
 
Theoremtailfb 35257 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    β‡’   ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ ran (tailβ€˜π·) ∈ (fBasβ€˜π‘‹))
 
Theoremfilnetlem1 35258* Lemma for filnet 35262. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)    &   π· = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}    &   π΄ ∈ V    &   π΅ ∈ V    β‡’   (𝐴𝐷𝐡 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐡 ∈ 𝐻) ∧ (1st β€˜π΅) βŠ† (1st β€˜π΄)))
 
Theoremfilnetlem2 35259* Lemma for filnet 35262. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)    &   π· = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}    β‡’   (( I β†Ύ 𝐻) βŠ† 𝐷 ∧ 𝐷 βŠ† (𝐻 Γ— 𝐻))
 
Theoremfilnetlem3 35260* Lemma for filnet 35262. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)    &   π· = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}    β‡’   (𝐻 = βˆͺ βˆͺ 𝐷 ∧ (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐻 βŠ† (𝐹 Γ— 𝑋) ∧ 𝐷 ∈ DirRel)))
 
Theoremfilnetlem4 35261* Lemma for filnet 35262. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)    &   π· = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}    β‡’   (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆƒπ‘‘ ∈ DirRel βˆƒπ‘“(𝑓:dom π‘‘βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘))))
 
Theoremfilnet 35262* 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 ∈ (Filβ€˜π‘‹) β†’ βˆƒπ‘‘ ∈ DirRel βˆƒπ‘“(𝑓:dom π‘‘βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘))))
 
21.13  Mathbox for Anthony Hart
 
21.13.1  Propositional Calculus
 
Theoremtb-ax1 35263 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.)
((πœ‘ β†’ πœ“) β†’ ((πœ“ β†’ πœ’) β†’ (πœ‘ β†’ πœ’)))
 
Theoremtb-ax2 35264 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.)
(πœ‘ β†’ (πœ“ β†’ πœ‘))
 
Theoremtb-ax3 35265 The third of three axioms in the Tarski-Bernays axiom system.

This axiom, along with ax-mp 5, tb-ax1 35263, and tb-ax2 35264, can be used to derive any theorem or rule that uses only β†’. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(((πœ‘ β†’ πœ“) β†’ πœ‘) β†’ πœ‘)
 
Theoremtbsyl 35266 The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ πœ“)    &   (πœ“ β†’ πœ’)    β‡’   (πœ‘ β†’ πœ’)
 
Theoremre1ax2lem 35267 Lemma for re1ax2 35268. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((πœ‘ β†’ (πœ“ β†’ πœ’)) β†’ (πœ“ β†’ (πœ‘ β†’ πœ’)))
 
Theoremre1ax2 35268 ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 35263 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.)
((πœ‘ β†’ (πœ“ β†’ πœ’)) β†’ ((πœ‘ β†’ πœ“) β†’ (πœ‘ β†’ πœ’)))
 
Theoremnaim1 35269 Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.)
((πœ‘ β†’ πœ“) β†’ ((πœ“ ⊼ πœ’) β†’ (πœ‘ ⊼ πœ’)))
 
Theoremnaim2 35270 Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.)
((πœ‘ β†’ πœ“) β†’ ((πœ’ ⊼ πœ“) β†’ (πœ’ ⊼ πœ‘)))
 
Theoremnaim1i 35271 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ β†’ πœ“)    &   (πœ“ ⊼ πœ’)    β‡’   (πœ‘ ⊼ πœ’)
 
Theoremnaim2i 35272 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ β†’ πœ“)    &   (πœ’ ⊼ πœ“)    β‡’   (πœ’ ⊼ πœ‘)
 
Theoremnaim12i 35273 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ β†’ πœ“)    &   (πœ’ β†’ πœƒ)    &   (πœ“ ⊼ πœƒ)    β‡’   (πœ‘ ⊼ πœ’)
 
Theoremnabi1i 35274 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ ↔ πœ“)    &   (πœ“ ⊼ πœ’)    β‡’   (πœ‘ ⊼ πœ’)
 
Theoremnabi2i 35275 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ ↔ πœ“)    &   (πœ’ ⊼ πœ“)    β‡’   (πœ’ ⊼ πœ‘)
 
Theoremnabi12i 35276 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ ↔ πœ“)    &   (πœ’ ↔ πœƒ)    &   (πœ“ ⊼ πœƒ)    β‡’   (πœ‘ ⊼ πœ’)
 
Syntaxw3nand 35277 The double nand.
wff (πœ‘ ⊼ πœ“ ⊼ πœ’)
 
Definitiondf-3nand 35278 The double nand. This definition allows to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.)
((πœ‘ ⊼ πœ“ ⊼ πœ’) ↔ (πœ‘ β†’ (πœ“ β†’ Β¬ πœ’)))
 
Theoremdf3nandALT1 35279 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((πœ‘ ⊼ πœ“ ⊼ πœ’) ↔ (πœ‘ ⊼ ((πœ“ ⊼ πœ’) ⊼ (πœ“ ⊼ πœ’))))
 
Theoremdf3nandALT2 35280 The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
((πœ‘ ⊼ πœ“ ⊼ πœ’) ↔ Β¬ (πœ‘ ∧ πœ“ ∧ πœ’))
 
Theoremandnand1 35281 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((πœ‘ ∧ πœ“ ∧ πœ’) ↔ ((πœ‘ ⊼ πœ“ ⊼ πœ’) ⊼ (πœ‘ ⊼ πœ“ ⊼ πœ’)))
 
Theoremimnand2 35282 An β†’ nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
((Β¬ πœ‘ β†’ πœ“) ↔ ((πœ‘ ⊼ πœ‘) ⊼ (πœ“ ⊼ πœ“)))
 
21.13.2  Predicate Calculus
 
Theoremnalfal 35283 Not all sets hold βŠ₯ as true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆ€π‘₯βŠ₯
 
Theoremnexntru 35284 There does not exist a set such that ⊀ is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒπ‘₯ Β¬ ⊀
 
Theoremnexfal 35285 There does not exist a set such that βŠ₯ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒπ‘₯βŠ₯
 
Theoremneufal 35286 There does not exist exactly one set such that βŠ₯ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒ!π‘₯βŠ₯
 
Theoremneutru 35287 There does not exist exactly one set such that ⊀ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒ!π‘₯⊀
 
Theoremnmotru 35288 There does not exist at most one set such that ⊀ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒ*π‘₯⊀
 
Theoremmofal 35289 There exist at most one set such that βŠ₯ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
βˆƒ*π‘₯βŠ₯
 
Theoremnrmo 35290 "At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.)
(π‘₯ ∈ 𝐴 β†’ Β¬ πœ‘)    β‡’   βˆƒ*π‘₯ ∈ 𝐴 πœ‘
 
21.13.3  Miscellaneous single axioms
 
Theoremmeran1 35291 A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(Β¬ (Β¬ (Β¬ πœ‘ ∨ πœ“) ∨ (πœ’ ∨ (πœƒ ∨ 𝜏))) ∨ (Β¬ (Β¬ πœƒ ∨ πœ‘) ∨ (πœ’ ∨ (𝜏 ∨ πœ‘))))
 
Theoremmeran2 35292 A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(Β¬ (Β¬ (Β¬ πœ‘ ∨ πœ“) ∨ (πœ’ ∨ (πœƒ ∨ 𝜏))) ∨ (Β¬ (Β¬ 𝜏 ∨ πœƒ) ∨ (πœ’ ∨ (πœ‘ ∨ πœƒ))))
 
Theoremmeran3 35293 A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(Β¬ (Β¬ (Β¬ πœ‘ ∨ πœ“) ∨ (πœ’ ∨ (πœƒ ∨ 𝜏))) ∨ (Β¬ (Β¬ πœ’ ∨ πœ‘) ∨ (𝜏 ∨ (πœƒ ∨ πœ‘))))
 
Theoremwaj-ax 35294 A single axiom for propositional calculus discovered by Mordchaj Wajsberg (Logical Works, Polish Academy of Sciences, 1977). See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom W on slide 8). (Contributed by Anthony Hart, 13-Aug-2011.)
((πœ‘ ⊼ (πœ“ ⊼ πœ’)) ⊼ (((πœƒ ⊼ πœ’) ⊼ ((πœ‘ ⊼ πœƒ) ⊼ (πœ‘ ⊼ πœƒ))) ⊼ (πœ‘ ⊼ (πœ‘ ⊼ πœ“))))
 
Theoremlukshef-ax2 35295 A single axiom for propositional calculus discovered by Jan Lukasiewicz. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom L2 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.)
((πœ‘ ⊼ (πœ“ ⊼ πœ’)) ⊼ ((πœ‘ ⊼ (πœ’ ⊼ πœ‘)) ⊼ ((πœƒ ⊼ πœ“) ⊼ ((πœ‘ ⊼ πœƒ) ⊼ (πœ‘ ⊼ πœƒ)))))
 
Theoremarg-ax 35296 A single axiom for propositional calculus discovered by Ken Harris and Branden Fitelson. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom HF1 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.)
((πœ‘ ⊼ (πœ“ ⊼ πœ’)) ⊼ ((πœ‘ ⊼ (πœ“ ⊼ πœ’)) ⊼ ((πœƒ ⊼ πœ’) ⊼ ((πœ’ ⊼ πœƒ) ⊼ (πœ‘ ⊼ πœƒ)))))
 
21.13.4  Connective Symmetry
 
Theoremnegsym1 35297 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 πœ‘ " means that "something is true of πœ‘". The expression "delta πœ‘ " can be substituted with Β¬ πœ‘, πœ“ ∧ πœ‘, βˆ€π‘₯πœ‘, etc.

Later on, Meredith discovered a single axiom, in the form of ( delta delta βŠ₯ β†’ delta πœ‘ ). 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 Β¬. (Contributed by Anthony Hart, 4-Sep-2011.)

(Β¬ Β¬ βŠ₯ β†’ Β¬ πœ‘)
 
Theoremimsym1 35298 A symmetry with β†’.

See negsym1 35297 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ β†’ (πœ“ β†’ βŠ₯)) β†’ (πœ“ β†’ πœ‘))
 
Theorembisym1 35299 A symmetry with ↔.

See negsym1 35297 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ ↔ (πœ“ ↔ βŠ₯)) β†’ (πœ“ ↔ πœ‘))
 
Theoremconsym1 35300 A symmetry with ∧.

See negsym1 35297 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ ∧ (πœ“ ∧ βŠ₯)) β†’ (πœ“ ∧ πœ‘))
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