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Theorem List for Metamath Proof Explorer - 31501-31600   *Has distinct variable group(s)
TypeLabelDescription
Statement

21.9.4  Refinements

Syntaxcfne 31501 Extend class definition to include the "finer than" relation.
class Fne

Definitiondf-fne 31502* Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.)
Fne = {⟨𝑥, 𝑦⟩ ∣ ( 𝑥 = 𝑦 ∧ ∀𝑧𝑥 𝑧 (𝑦 ∩ 𝒫 𝑧))}

Theoremfnerel 31503 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
Rel Fne

Theoremisfne 31504* 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 31505 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 31506 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 31507* 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 31508* 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 31509 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐴Fne𝐵𝑋 = 𝑌)

Theoremfnetg 31510 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝐴Fne𝐵𝐴 ⊆ (topGen‘𝐵))

Theoremfnessex 31511* 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 31512* 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 31513* 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 31514 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
𝑋 = 𝐴    &   𝑌 = 𝐵       ((𝐵𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Fne𝐵)

Theoremfneref 31515 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
(𝐴𝑉𝐴Fne𝐴)

Theoremfnetr 31516 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
((𝐴Fne𝐵𝐵Fne𝐶) → 𝐴Fne𝐶)

Theoremfneval 31517 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 31518 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 31519 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 31520 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 31521 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 31522* 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 31523* 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))))

21.9.5  Neighborhood bases determine topologies

Theoremneibastop1 31524* 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 31525* Lemma for neibastop2 31526. (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 31526* 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 31527* 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) ≠ ∅})

21.9.6  Lattice structure of topologies

Theoremtopmtcl 31528 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 31529* 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 31530* 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 31531* 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 31532* 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 31533* 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 31534* 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)))

21.9.7  Filter bases

Theoremfgmin 31535 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 31536* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 21456. (Contributed by Jeff Hankins, 3-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → (𝑋filGen{𝑥𝐽𝑆𝑥}) = ((nei‘𝐽)‘𝑆))

21.9.8  Directed sets, nets

Theoremtailfval 31537* 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 31538 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 31539 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 31540 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 31541 A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)
𝑋 = dom 𝐷       ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → 𝐴 ∈ ((tail‘𝐷)‘𝐴))

Theoremtailfb 31542 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 31543* Lemma for filnet 31547. 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 31544* Lemma for filnet 31547. 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 31545* Lemma for filnet 31547. (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 31546* Lemma for filnet 31547. (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 31547* 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))))

21.10  Mathbox for Anthony Hart

21.10.1  Propositional Calculus

Theoremtb-ax1 31548 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 31549 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 31550 The third of three axioms in the Tarski-Bernays axiom system.

This axiom, along with ax-mp 5, tb-ax1 31548, and tb-ax2 31549, 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 31551 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 31552 Lemma for re1ax2 31553. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremre1ax2 31553 ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 31548 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 31554 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremnaim2 31555 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜒𝜓) → (𝜒𝜑)))

Theoremnaim1i 31556 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremnaim2i 31557 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜒𝜑)

Theoremnaim12i 31558 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓𝜃)       (𝜑𝜒)

Theoremnabi1 31559 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremnabi2 31560 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Theoremnabi1i 31561 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremnabi2i 31562 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜒𝜑)

Theoremnabi12i 31563 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓𝜃)       (𝜑𝜒)

Syntaxw3nand 31564 The double nand.
wff (𝜑𝜓𝜒)

Definitiondf-3nand 31565 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 31566 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((𝜑𝜓𝜒) ↔ (𝜑 ⊼ ((𝜓𝜒) ⊼ (𝜓𝜒))))

Theoremdf3nandALT2 31567 The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
((𝜑𝜓𝜒) ↔ ¬ (𝜑𝜓𝜒))

Theoremandnand1 31568 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓𝜒) ⊼ (𝜑𝜓𝜒)))

Theoremimnand2 31569 An nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
((¬ 𝜑𝜓) ↔ ((𝜑𝜑) ⊼ (𝜓𝜓)))

21.10.2  Predicate Calculus

Theoremallt 31570 For all sets, is true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥

Theoremalnof 31571 For all sets, is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥 ¬ ⊥

Theoremnalf 31572 Not all sets hold as true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∀𝑥

Theoremextt 31573 There exists a set that holds as true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥

Theoremnextnt 31574 There does not exist a set, such that is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃𝑥 ¬ ⊤

Theoremnextf 31575 There does not exist a set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃𝑥

Theoremunnf 31576 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃!𝑥

Theoremunnt 31577 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃!𝑥

Theoremmont 31578 There does not exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃*𝑥

Theoremmof 31579 There exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
∃*𝑥

21.10.3  Misc. Single Axiom Systems

Theoremmeran1 31580 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))

Theoremmeran2 31581 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜏𝜃) ∨ (𝜒 ∨ (𝜑𝜃))))

Theoremmeran3 31582 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜒𝜑) ∨ (𝜏 ∨ (𝜃𝜑))))

Theoremwaj-ax 31583 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 31584 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 31585 ? (Contributed by Anthony Hart, 14-Aug-2011.)
((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜃𝜒) ⊼ ((𝜒𝜃) ⊼ (𝜑𝜃)))))

21.10.4  Connective Symmetry

Theoremnegsym1 31586 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)

Theoremimsym1 31587 A symmetry with .

((𝜓 → (𝜓 → ⊥)) → (𝜓𝜑))

Theorembisym1 31588 A symmetry with .

((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓𝜑))

Theoremconsym1 31589 A symmetry with .

((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓𝜑))

Theoremdissym1 31590 A symmetry with .

((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓𝜑))

Theoremnandsym1 31591 A symmetry with .

((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓𝜑))

Theoremunisym1 31592 A symmetry with .

See negsym1 31586 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

(∀𝑥𝑥⊥ → ∀𝑥𝜑)

Theoremexisym1 31593 A symmetry with .

(∃𝑥𝑥⊥ → ∃𝑥𝜑)

Theoremunqsym1 31594 A symmetry with ∃!.

(∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)

Theoremamosym1 31595 A symmetry with ∃*.

(∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Theoremsubsym1 31596 A symmetry with [𝑥 / 𝑦].

([𝑥 / 𝑦][𝑥 / 𝑦]⊥ → [𝑥 / 𝑦]𝜑)

21.11  Mathbox for Chen-Pang He

21.11.1  Ordinal topology

Theoremontopbas 31597 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
(𝐵 ∈ On → 𝐵 ∈ TopBases)

Theoremonsstopbas 31598 The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.)
On ⊆ TopBases

Theoremonpsstopbas 31599 The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.)
On ⊊ TopBases

Theoremontgval 31600 The topology generated from an ordinal number 𝐵 is suc 𝐵. (Contributed by Chen-Pang He, 10-Oct-2015.)
(𝐵 ∈ On → (topGen‘𝐵) = suc 𝐵)

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