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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | isfne4 33801 | 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 33802 | 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 33803* | 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 33804* | 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 33805 | 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 33806 | A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝐴Fne𝐵 → 𝐴 ⊆ (topGen‘𝐵)) | ||
Theorem | fnessex 33807* | 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 33808* | 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 33809* | 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 33810 | 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 33811 | Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴Fne𝐴) | ||
Theorem | fnetr 33812 | 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 33813 | 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 33814 | 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 33815 | 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 33816 | 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 33817 | 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 33818* | 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 33819* | 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 33820* | 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 33821* | Lemma for neibastop2 33822. (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 33822* | 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 33823* | 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 33824 | 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 33825* | 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 33826* | 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 33827* | 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 33828* | 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 33829* | 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 33830* | 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 33831 | 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 33832* | The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 22447. (Contributed by Jeff Hankins, 3-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋filGen{𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) = ((nei‘𝐽)‘𝑆)) | ||
Theorem | tailfval 33833* | 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 33834 | 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 33835 | 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 33836 | 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 33837 | 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 33838 | 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 33839* | Lemma for filnet 33843. 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 33840* | Lemma for filnet 33843. 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 33841* | Lemma for filnet 33843. (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 33842* | Lemma for filnet 33843. (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 33843* | 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 33844 | 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 33845 | 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 33846 |
The third of three axioms in the Tarski-Bernays axiom system.
This axiom, along with ax-mp 5, tb-ax1 33844, and tb-ax2 33845, 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 33847 | 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 33848 | Lemma for re1ax2 33849. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | re1ax2 33849 | ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 33844 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 33850 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒))) | ||
Theorem | naim2 33851 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑))) | ||
Theorem | naim1i 33852 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 ⊼ 𝜒) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | naim2i 33853 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 ⊼ 𝜓) ⇒ ⊢ (𝜒 ⊼ 𝜑) | ||
Theorem | naim12i 33854 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ (𝜓 ⊼ 𝜃) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | nabi1i 33855 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜓 ⊼ 𝜒) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | nabi2i 33856 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ⊼ 𝜓) ⇒ ⊢ (𝜒 ⊼ 𝜑) | ||
Theorem | nabi12i 33857 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) & ⊢ (𝜓 ⊼ 𝜃) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Syntax | w3nand 33858 | The double nand. |
wff (𝜑 ⊼ 𝜓 ⊼ 𝜒) | ||
Definition | df-3nand 33859 | 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 33860 | 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 33861 | 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 33862 | 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 33863 | An → nand relation. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) | ||
Theorem | nalfal 33864 | Not all sets hold ⊥ as true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∀𝑥⊥ | ||
Theorem | nexntru 33865 | There does not exist a set such that ⊤ is not true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃𝑥 ¬ ⊤ | ||
Theorem | nexfal 33866 | There does not exist a set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃𝑥⊥ | ||
Theorem | neufal 33867 | There does not exist exactly one set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃!𝑥⊥ | ||
Theorem | neutru 33868 | There does not exist exactly one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃!𝑥⊤ | ||
Theorem | nmotru 33869 | There does not exist at most one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃*𝑥⊤ | ||
Theorem | mofal 33870 | There exist at most one set such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ∃*𝑥⊥ | ||
Theorem | nrmo 33871 | "At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
⊢ (𝑥 ∈ 𝐴 → ¬ 𝜑) ⇒ ⊢ ∃*𝑥 ∈ 𝐴 𝜑 | ||
Theorem | meran1 33872 | A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜃 ∨ 𝜑) ∨ (𝜒 ∨ (𝜏 ∨ 𝜑)))) | ||
Theorem | meran2 33873 | A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜏 ∨ 𝜃) ∨ (𝜒 ∨ (𝜑 ∨ 𝜃)))) | ||
Theorem | meran3 33874 | A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑)))) | ||
Theorem | waj-ax 33875 | 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.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ (((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))) ⊼ (𝜑 ⊼ (𝜑 ⊼ 𝜓)))) | ||
Theorem | lukshef-ax2 33876 | 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.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜑)) ⊼ ((𝜃 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | arg-ax 33877 | 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.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜒 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | negsym1 33878 |
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". The expression "delta
𝜑
" can be
substituted with ¬ 𝜑, 𝜓 ∧ 𝜑, ∀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) | ||
Theorem | imsym1 33879 |
A symmetry with →.
See negsym1 33878 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 → (𝜓 → ⊥)) → (𝜓 → 𝜑)) | ||
Theorem | bisym1 33880 |
A symmetry with ↔.
See negsym1 33878 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓 ↔ 𝜑)) | ||
Theorem | consym1 33881 |
A symmetry with ∧.
See negsym1 33878 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑)) | ||
Theorem | dissym1 33882 |
A symmetry with ∨.
See negsym1 33878 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓 ∨ 𝜑)) | ||
Theorem | nandsym1 33883 |
A symmetry with ⊼.
See negsym1 33878 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑)) | ||
Theorem | unisym1 33884 |
A symmetry with ∀.
See negsym1 33878 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
⊢ (∀𝑥∀𝑥⊥ → ∀𝑥𝜑) | ||
Theorem | exisym1 33885 |
A symmetry with ∃.
See negsym1 33878 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ (∃𝑥∃𝑥⊥ → ∃𝑥𝜑) | ||
Theorem | unqsym1 33886 |
A symmetry with ∃!.
See negsym1 33878 for more information. (Contributed by Anthony Hart, 6-Sep-2011.) |
⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑) | ||
Theorem | amosym1 33887 |
A symmetry with ∃*.
See negsym1 33878 for more information. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑) | ||
Theorem | subsym1 33888 |
A symmetry with [𝑥 / 𝑦].
See negsym1 33878 for more information. (Contributed by Anthony Hart, 11-Sep-2011.) |
⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑) | ||
Theorem | ontopbas 33889 | An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.) |
⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases) | ||
Theorem | onsstopbas 33890 | The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.) |
⊢ On ⊆ TopBases | ||
Theorem | onpsstopbas 33891 | 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 | ||
Theorem | ontgval 33892 | 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) | ||
Theorem | ontgsucval 33893 | The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.) |
⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) | ||
Theorem | onsuctop 33894 | A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top) | ||
Theorem | onsuctopon 33895 | One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴)) | ||
Theorem | ordtoplem 33896 | Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆) ⇒ ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆)) | ||
Theorem | ordtop 33897 | An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) | ||
Theorem | onsucconni 33898 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
⊢ 𝐴 ∈ On ⇒ ⊢ suc 𝐴 ∈ Conn | ||
Theorem | onsucconn 33899 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn) | ||
Theorem | ordtopconn 33900 | An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn)) |
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