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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cfne 31501 | Extend class definition to include the "finer than" relation. |
class Fne | ||
Definition | df-fne 31502* | 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 31503 | Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.) |
⊢ Rel Fne | ||
Theorem | isfne 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 ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) | ||
Theorem | isfne4 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 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) | ||
Theorem | isfne4b 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 ⇒ ⊢ (𝐵 ∈ 𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)))) | ||
Theorem | isfne2 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 ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))) | ||
Theorem | isfne3 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 ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))) | ||
Theorem | fnebas 31509 | 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 31510 | A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝐴Fne𝐵 → 𝐴 ⊆ (topGen‘𝐵)) | ||
Theorem | fnessex 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.) |
⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝑆) → ∃𝑥 ∈ 𝐵 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑆)) | ||
Theorem | fneuni 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.) |
⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥)) | ||
Theorem | fneint 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 → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}) | ||
Theorem | fness 31514 | 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 31515 | Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴Fne𝐴) | ||
Theorem | fnetr 31516 | 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 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) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∼ 𝐵 ↔ (topGen‘𝐴) = (topGen‘𝐵))) | ||
Theorem | fneer 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 | ||
Theorem | topfne 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) → (𝐽 ⊆ 𝐾 ↔ 𝐽Fne𝐾)) | ||
Theorem | topfneec 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)) | ||
Theorem | topfneec2 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)) | ||
Theorem | fnessref 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(𝑐 ⊆ 𝐵 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)))) | ||
Theorem | refssfne 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(𝐵 ⊆ 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴)))) | ||
Theorem | neibastop1 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.) |
⊢ (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 31525* | Lemma for neibastop2 31526. (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 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.) |
⊢ (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 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.) |
⊢ (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 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋)) | ||
Theorem | topmeet 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) = ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗}) | ||
Theorem | topjoin 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ ∪ 𝑆))) = ∩ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑗 ⊆ 𝑘}) | ||
Theorem | fnemeet1 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝐴) | ||
Theorem | fnemeet2 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥))) | ||
Theorem | fnejoin1 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆)) | ||
Theorem | fnejoin2 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇))) | ||
Theorem | fgmin 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)) → (𝐵 ⊆ 𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹)) | ||
Theorem | neifg 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 ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋filGen{𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) = ((nei‘𝐽)‘𝑆)) | ||
Theorem | tailfval 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) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) | ||
Theorem | tailval 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 ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴})) | ||
Theorem | eltail 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 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝐶) → (𝐵 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐵)) | ||
Theorem | tailf 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) | ||
Theorem | tailini 31541 | 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 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)) | ||
Theorem | filnetlem1 31543* | Lemma for filnet 31547. 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 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 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) & ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} ⇒ ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻)) | ||
Theorem | filnetlem3 31545* | Lemma for filnet 31547. (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 31546* | Lemma for filnet 31547. (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 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 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)))) | ||
Theorem | tb-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.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | tb-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 → (𝜓 → 𝜑)) | ||
Theorem | tb-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.) |
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) | ||
Theorem | tbsyl 31551 | 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 31552 | Lemma for re1ax2 31553. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | re1ax2 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 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | naim1 31554 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒))) | ||
Theorem | naim2 31555 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑))) | ||
Theorem | naim1i 31556 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 ⊼ 𝜒) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | naim2i 31557 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 ⊼ 𝜓) ⇒ ⊢ (𝜒 ⊼ 𝜑) | ||
Theorem | naim12i 31558 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ (𝜓 ⊼ 𝜃) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | nabi1 31559 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒))) | ||
Theorem | nabi2 31560 | Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓))) | ||
Theorem | nabi1i 31561 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜓 ⊼ 𝜒) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Theorem | nabi2i 31562 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ⊼ 𝜓) ⇒ ⊢ (𝜒 ⊼ 𝜑) | ||
Theorem | nabi12i 31563 | Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) & ⊢ (𝜓 ⊼ 𝜃) ⇒ ⊢ (𝜑 ⊼ 𝜒) | ||
Syntax | w3nand 31564 | The double nand. |
wff (𝜑 ⊼ 𝜓 ⊼ 𝜒) | ||
Definition | df-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.) |
⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒))) | ||
Theorem | df3nandALT1 31566 | 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 31567 | 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 31568 | 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 31569 | An → nand relation. (Contributed by Anthony Hart, 2-Sep-2011.) |
⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) | ||
Theorem | allt 31570 | For all sets, ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ∀𝑥⊤ | ||
Theorem | alnof 31571 | For all sets, ⊥ is not true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ∀𝑥 ¬ ⊥ | ||
Theorem | nalf 31572 | Not all sets hold ⊥ as true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∀𝑥⊥ | ||
Theorem | extt 31573 | There exists a set that holds ⊤ as true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ∃𝑥⊤ | ||
Theorem | nextnt 31574 | There does not exist a set, such that ⊤ is not true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃𝑥 ¬ ⊤ | ||
Theorem | nextf 31575 | There does not exist a set, such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃𝑥⊥ | ||
Theorem | unnf 31576 | There does not exist exactly one set, such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃!𝑥⊥ | ||
Theorem | unnt 31577 | There does not exist exactly one set, such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃!𝑥⊤ | ||
Theorem | mont 31578 | There does not exist at most one set, such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ¬ ∃*𝑥⊤ | ||
Theorem | mof 31579 | There exist at most one set, such that ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ ∃*𝑥⊥ | ||
Theorem | meran1 31580 | A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜃 ∨ 𝜑) ∨ (𝜒 ∨ (𝜏 ∨ 𝜑)))) | ||
Theorem | meran2 31581 | A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜏 ∨ 𝜃) ∨ (𝜒 ∨ (𝜑 ∨ 𝜃)))) | ||
Theorem | meran3 31582 | A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑)))) | ||
Theorem | waj-ax 31583 | A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ (((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))) ⊼ (𝜑 ⊼ (𝜑 ⊼ 𝜓)))) | ||
Theorem | lukshef-ax2 31584 | A single axiom for propositional calculus offered by Lukasiewicz. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜑)) ⊼ ((𝜃 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | arg-ax 31585 | ? (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜒 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | negsym1 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, 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 31587 |
A symmetry with →.
See negsym1 31586 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 → (𝜓 → ⊥)) → (𝜓 → 𝜑)) | ||
Theorem | bisym1 31588 |
A symmetry with ↔.
See negsym1 31586 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓 ↔ 𝜑)) | ||
Theorem | consym1 31589 |
A symmetry with ∧.
See negsym1 31586 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑)) | ||
Theorem | dissym1 31590 |
A symmetry with ∨.
See negsym1 31586 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓 ∨ 𝜑)) | ||
Theorem | nandsym1 31591 |
A symmetry with ⊼.
See negsym1 31586 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑)) | ||
Theorem | unisym1 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.) |
⊢ (∀𝑥∀𝑥⊥ → ∀𝑥𝜑) | ||
Theorem | exisym1 31593 |
A symmetry with ∃.
See negsym1 31586 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
⊢ (∃𝑥∃𝑥⊥ → ∃𝑥𝜑) | ||
Theorem | unqsym1 31594 |
A symmetry with ∃!.
See negsym1 31586 for more information. (Contributed by Anthony Hart, 6-Sep-2011.) |
⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑) | ||
Theorem | amosym1 31595 |
A symmetry with ∃*.
See negsym1 31586 for more information. (Contributed by Anthony Hart, 13-Sep-2011.) |
⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑) | ||
Theorem | subsym1 31596 |
A symmetry with [𝑥 / 𝑦].
See negsym1 31586 for more information. (Contributed by Anthony Hart, 11-Sep-2011.) |
⊢ ([𝑥 / 𝑦][𝑥 / 𝑦]⊥ → [𝑥 / 𝑦]𝜑) | ||
Theorem | ontopbas 31597 | An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.) |
⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases) | ||
Theorem | onsstopbas 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 | ||
Theorem | onpsstopbas 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 | ||
Theorem | ontgval 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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