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Theorem List for Metamath Proof Explorer - 45701-45800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitsclinecirc0b 45701 The intersection points of a line through two different points and a circle around the origin, using the definition of a line in a two dimensional Euclidean space. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   𝐿 = (LineM𝐸)    &   𝐴 = ((𝑋‘2) − (𝑌‘2))    &   𝐵 = ((𝑌‘1) − (𝑋‘1))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑍 ∈ ( 0 𝑆𝑅) ∧ 𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍𝑃 ∧ (((𝑍‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑍‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑍‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑍‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))))
 
Theoremitsclinecirc0in 45702 The intersection points of a line through two different points and a circle around the origin, using the definition of a line in a two dimensional Euclidean space, expressed as intersection. (Contributed by AV, 7-May-2023.) (Revised by AV, 14-May-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   𝐿 = (LineM𝐸)    &   𝐴 = ((𝑋‘2) − (𝑌‘2))    &   𝐵 = ((𝑌‘1) − (𝑋‘1))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) = {{⟨1, (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄)⟩, ⟨2, (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)⟩}, {⟨1, (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄)⟩, ⟨2, (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)⟩}})
 
Theoremitsclquadb 45703* Quadratic equation for the y-coordinate of the intersection points of a line and a circle. (Contributed by AV, 22-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))       ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+𝑌 ∈ ℝ) → (∃𝑥 ∈ ℝ (((𝑥↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑥) + (𝐵 · 𝑌)) = 𝐶) ↔ ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
 
Theoremitsclquadeu 45704* Quadratic equation for the y-coordinate of the intersection points of a line and a circle. (Contributed by AV, 23-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))       ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+𝑌 ∈ ℝ) → (∃!𝑥 ∈ ℝ (((𝑥↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑥) + (𝐵 · 𝑌)) = 𝐶) ↔ ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
 
Theorem2itscplem1 45705 Lemma 1 for 2itscp 45708. (Contributed by AV, 4-Mar-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)       (𝜑 → ((((𝐸↑2) · (𝐵↑2)) + ((𝐷↑2) · (𝐴↑2))) − (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) = (((𝐷 · 𝐴) − (𝐸 · 𝐵))↑2))
 
Theorem2itscplem2 45706 Lemma 2 for 2itscp 45708. (Contributed by AV, 4-Mar-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))       (𝜑 → (𝐶↑2) = ((((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) + ((𝐸↑2) · (𝐴↑2))))
 
Theorem2itscplem3 45707 Lemma D for 2itscp 45708. (Contributed by AV, 4-Mar-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   𝑄 = ((𝐸↑2) + (𝐷↑2))    &   𝑆 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (𝜑𝑆 = ((((𝐸↑2) · ((𝑅↑2) − (𝐴↑2))) + ((𝐷↑2) · ((𝑅↑2) − (𝐵↑2)))) − (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))))
 
Theorem2itscp 45708 A condition for a quadratic equation with real coefficients (for the intersection points of a line with a circle) to have (exactly) two different real solutions. (Contributed by AV, 5-Mar-2023.) (Revised by AV, 16-May-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))    &   (𝜑 → (𝐵𝑌𝐴𝑋))    &   𝑄 = ((𝐸↑2) + (𝐷↑2))    &   𝑆 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (𝜑 → 0 < 𝑆)
 
Theoremitscnhlinecirc02plem1 45709 Lemma 1 for itscnhlinecirc02p 45712. (Contributed by AV, 6-Mar-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))    &   (𝜑𝐵𝑌)       (𝜑 → 0 < ((-(2 · (𝐷 · 𝐶))↑2) − (4 · (((𝐸↑2) + (𝐷↑2)) · ((𝐶↑2) − ((𝐸↑2) · (𝑅↑2)))))))
 
Theoremitscnhlinecirc02plem2 45710 Lemma 2 for itscnhlinecirc02p 45712. (Contributed by AV, 10-Mar-2023.)
𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐵 · 𝑋) − (𝐴 · 𝑌))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) ∧ 𝐵𝑌) ∧ (𝑅 ∈ ℝ ∧ ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))) → 0 < ((-(2 · (𝐷 · 𝐶))↑2) − (4 · (((𝐸↑2) + (𝐷↑2)) · ((𝐶↑2) − ((𝐸↑2) · (𝑅↑2)))))))
 
Theoremitscnhlinecirc02plem3 45711 Lemma 3 for itscnhlinecirc02p 45712. (Contributed by AV, 10-Mar-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝐿 = (LineM𝐸)    &   𝐷 = (dist‘𝐸)       (((𝑋𝑃𝑌𝑃 ∧ (𝑋‘2) ≠ (𝑌‘2)) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → 0 < ((-(2 · (((𝑌‘1) − (𝑋‘1)) · (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))))↑2) − (4 · (((((𝑋‘2) − (𝑌‘2))↑2) + (((𝑌‘1) − (𝑋‘1))↑2)) · (((((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))↑2) − ((((𝑋‘2) − (𝑌‘2))↑2) · (𝑅↑2)))))))
 
Theoremitscnhlinecirc02p 45712* Intersection of a nonhorizontal line with a circle: A nonhorizontal line passing through a point within a circle around the origin intersects the circle at exactly two different points. (Contributed by AV, 28-Jan-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝐿 = (LineM𝐸)    &   𝐷 = (dist‘𝐸)    &   𝑍 = {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}       (((𝑋𝑃𝑌𝑃 ∧ (𝑋‘2) ≠ (𝑌‘2)) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑠 ∈ 𝒫 ℝ((♯‘𝑠) = 2 ∧ ∀𝑦𝑠 ∃!𝑥 ∈ ℝ (𝑍 ∈ ( 0 𝑆𝑅) ∧ 𝑍 ∈ (𝑋𝐿𝑌))))
 
Theoreminlinecirc02plem 45713* Lemma for inlinecirc02p 45714. (Contributed by AV, 7-May-2023.) (Revised by AV, 15-May-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝐿 = (LineM𝐸)    &   𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   𝐴 = ((𝑋‘2) − (𝑌‘2))    &   𝐵 = ((𝑌‘1) − (𝑋‘1))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 < 𝐷)) → ∃𝑎𝑃𝑏𝑃 ((( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) = {𝑎, 𝑏} ∧ 𝑎𝑏))
 
Theoreminlinecirc02p 45714 Intersection of a line with a circle: A line passing through a point within a circle around the origin intersects the circle at exactly two different points. (Contributed by AV, 9-May-2023.) (Revised by AV, 16-May-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝐿 = (LineM𝐸)    &   𝐷 = (dist‘𝐸)       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) ∈ (Pairsproper𝑃))
 
Theoreminlinecirc02preu 45715* Intersection of a line with a circle: A line passing through a point within a circle around the origin intersects the circle at exactly two different points, expressed with restricted uniqueness (and without the definition of proper pairs). (Contributed by AV, 16-May-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝐿 = (LineM𝐸)    &   𝐷 = (dist‘𝐸)       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑝 ∈ 𝒫 𝑃((♯‘𝑝) = 2 ∧ 𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌))))
 
20.42  Mathbox for Zhi Wang
 
20.42.1  Propositional calculus
 
Theorempm4.71da 45716 Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 565. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 ↔ (𝜓𝜒)))
 
Theoremlogic1 45717 Distribution of implication over biconditional with replacement (deduction form). (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremlogic1a 45718 Variant of logic1 45717. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓𝜒))    &   ((𝜑𝜓) → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremlogic2 45719 Variant of logic1 45717. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ((𝜓𝜒) → (𝜃𝜏)))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theorempm5.32dav 45720 Distribution of implication over biconditional (deduction form). Variant of pm5.32da 582. (Contributed by Zhi Wang, 30-Aug-2024.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜓)))
 
Theorempm5.32dra 45721 Reverse distribution of implication over biconditional (deduction form). (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))       ((𝜑𝜓) → (𝜒𝜃))
 
Theoremexp12bd 45722 The import-export theorem (impexp 454) for biconditionals (deduction form). (Contributed by Zhi Wang, 3-Sep-2024.)
(𝜑 → (((𝜓𝜒) → 𝜃) ↔ ((𝜏𝜂) → 𝜁)))       (𝜑 → ((𝜓 → (𝜒𝜃)) ↔ (𝜏 → (𝜂𝜁))))
 
Theoremmonepilem 45723 Common lemmas for proving monomorphisms, epimorphisms, and potentially others. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))    &   ((𝜑𝜒) → 𝜃)       (𝜑 → (𝜓𝜒))
 
20.42.2  Predicate calculus with equality
 
20.42.2.1  Axiom scheme ax-5 (Distinctness)
 
Theoremdtrucor3 45724* An example of how ax-5 1917 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5247 in the ZF set theory. axc16nf 2264 and euae 2663 demonstrate that the violation of dtru 5247 leads to a model with only one object assuming its existence (ax-6 1975). The conclusion is also provable in the empty model ( see emptyal 1915). See also nf5 2287 and nf5i 2150 for the relation between unconditional ax-5 1917 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.)
¬ ∀𝑥 𝑥 = 𝑦    &   (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)       𝑥 𝑥 = 𝑦
 
20.42.3  ZF Set Theory - start with the Axiom of Extensionality
 
20.42.3.1  Restricted quantification
 
Theoremralbidb 45725* Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 45726 for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → (𝑥𝐴 ↔ (𝑥𝐵𝜓)))    &   ((𝜑𝑥𝐴) → (𝜒𝜃))       (𝜑 → (∀𝑥𝐴 𝜒 ↔ ∀𝑥𝐵 (𝜓𝜃)))
 
Theoremralbidc 45726* Formula-building rule for restricted universal quantifier and additional condition (deduction form). A variant of ralbidb 45725. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝑥𝐴 ↔ (𝑥𝐵𝜓)))    &   (𝜑 → ((𝑥𝐴 ∧ (𝑥𝐵𝜓)) → (𝜒𝜃)))       (𝜑 → (∀𝑥𝐴 𝜒 ↔ ∀𝑥𝐵 (𝜓𝜃)))
 
Theoremr19.41dv 45727* A complex deduction form of r19.41v 3252. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → ∃𝑥𝐴 𝜓)       ((𝜑𝜒) → ∃𝑥𝐴 (𝜓𝜒))
 
Theoremrextru 45728 Two ways of expressing "at least one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃𝑥 𝑥𝐴 ↔ ∃𝑥𝐴 ⊤)
 
Theoremrmotru 45729 Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by BJ, 23-Sep-2024.)
(∃*𝑥 𝑥𝐴 ↔ ∃*𝑥𝐴 ⊤)
 
Theoremreutru 45730 Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃!𝑥 𝑥𝐴 ↔ ∃!𝑥𝐴 ⊤)
 
TheoremreutruALT 45731 Alternate proof for reutru 45730. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃!𝑥 𝑥𝐴 ↔ ∃!𝑥𝐴 ⊤)
 
20.42.3.2  The empty set
 
Theoremssdisjd 45732 Subset preserves disjointness. Deduction form of ssdisj 4359. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑𝐴𝐵)    &   (𝜑 → (𝐵𝐶) = ∅)       (𝜑 → (𝐴𝐶) = ∅)
 
Theoremssdisjdr 45733 Subset preserves disjointness. Deduction form of ssdisj 4359. Alternatively this could be proved with ineqcom 4103 in tandem with ssdisjd 45732. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑𝐴𝐵)    &   (𝜑 → (𝐶𝐵) = ∅)       (𝜑 → (𝐶𝐴) = ∅)
 
Theoremdisjdifb 45734 Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐴𝐵) ∩ (𝐵𝐴)) = ∅
 
Theorempredisj 45735 Preimages of disjoint sets are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → Fun 𝐹)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑆 ⊆ (𝐹𝐴))    &   (𝜑𝑇 ⊆ (𝐹𝐵))       (𝜑 → (𝑆𝑇) = ∅)
 
20.42.3.3  Unordered and ordered pairs
 
Theoremvsn 45736 The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
{V} = ∅
 
Theoremmosn 45737* "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐴 = {𝐵} → ∃*𝑥 𝑥𝐴)
 
Theoremmo0 45738* "At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐴 = ∅ → ∃*𝑥 𝑥𝐴)
 
Theoremmosssn 45739* "At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.)
(𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥𝐴)
 
Theoremmo0sn 45740* Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 19-Sep-2024.)
(∃*𝑥 𝑥𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
 
Theoremmosssn2 45741* Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃*𝑥 𝑥𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
 
Theoremmof0 45742 There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
∃*𝑓 𝑓:𝐴⟶∅
 
Theoremmof02 45743* A variant of mof0 45742. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝐵 = ∅ → ∃*𝑓 𝑓:𝐴𝐵)
 
Theoremmof0ALT 45744* Alternate proof for mof0 45742 with stronger requirements on distinct variables. Uses mo4 2567. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
∃*𝑓 𝑓:𝐴⟶∅
 
Theoremeufsnlem 45745* There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 45746 assuming ax-rep 5164, or eufsn2 45747 assuming ax-pow 5242 and ax-un 7491. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑𝐵𝑊)    &   (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)       (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
Theoremeufsn 45746* There is exactly one function into a singleton, assuming ax-rep 5164. See eufsn2 45747 for different axiom requirements. If existence is not needed, use mofsn 45748 or mofsn2 45749 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑𝐵𝑊)    &   (𝜑𝐴𝑉)       (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
Theoremeufsn2 45747* There is exactly one function into a singleton, assuming ax-pow 5242 and ax-un 7491. Variant of eufsn 45746. If existence is not needed, use mofsn 45748 or mofsn2 45749 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑𝐵𝑊)    &   (𝜑𝐴𝑉)       (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
Theoremmofsn 45748* There is at most one function into a singleton, with fewer axioms than eufsn 45746 and eufsn2 45747. See also mofsn2 45749. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐵𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
 
Theoremmofsn2 45749* There is at most one function into a singleton. An unconditional variant of mofsn 45748, i.e., the singleton could be empty if 𝑌 is a proper class. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴𝐵)
 
Theoremmofsssn 45750* There is at most one function into a subclass of a singleton. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝐵 ⊆ {𝑌} → ∃*𝑓 𝑓:𝐴𝐵)
 
Theoremmofmo 45751* There is at most one function into a class containing at most one element. (Contributed by Zhi Wang, 19-Sep-2024.)
(∃*𝑥 𝑥𝐵 → ∃*𝑓 𝑓:𝐴𝐵)
 
20.42.4  ZF Set Theory - add the Axiom of Power Sets
 
20.42.4.1  Functions
 
Theoremelfvne0 45752 If a function value has a member, then the function is not an empty set (An artifact of our function value definition.) (Contributed by Zhi Wang, 16-Sep-2024.)
(𝐴 ∈ (𝐹𝐵) → 𝐹 ≠ ∅)
 
20.42.4.2  Operations
 
Theoremfvconstr 45753 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑𝐹 = (𝑅 × {𝑌}))    &   (𝜑𝑌𝑉)    &   (𝜑𝑌 ≠ ∅)       (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))
 
Theoremfvconstrn0 45754 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑𝐹 = (𝑅 × {𝑌}))    &   (𝜑𝑌𝑉)    &   (𝜑𝑌 ≠ ∅)       (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) ≠ ∅))
 
Theoremfvconstr2 45755 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑𝐹 = (𝑅 × {𝑌}))    &   (𝜑𝑋 ∈ (𝐴𝐹𝐵))       (𝜑𝐴𝑅𝐵)
 
20.42.5  ZF Set Theory - add the Axiom of Union
 
20.42.5.1  Equinumerosity
 
Theoremfvconst0ci 45756 A constant function's value is either the constant or the empty set. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
𝐵 ∈ V    &   𝑌 = ((𝐴 × {𝐵})‘𝑋)       (𝑌 = ∅ ∨ 𝑌 = 𝐵)
 
Theoremfvconstdomi 45757 A constant function's value is dominated by the constant. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
𝐵 ∈ V       ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵
 
Theoremf1omo 45758* There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 45757 assuming ax-un 7491 (see f1omoALT 45759). (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑𝐹 = (𝐴 × {1o}))       (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
 
Theoremf1omoALT 45759* There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Use f1omo 45758 without assuming ax-un 7491. (Contributed by Zhi Wang, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐹 = (𝐴 × {1o}))       (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
 
20.42.6  Order sets
 
20.42.6.1  Real number intervals
 
Theoremiccin 45760 Intersection of two closed intervals of extended reals. (Contributed by Zhi Wang, 9-Sep-2024.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = (if(𝐴𝐶, 𝐶, 𝐴)[,]if(𝐵𝐷, 𝐵, 𝐷)))
 
Theoremiccdisj2 45761 If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
((𝐴 ∈ ℝ*𝐷 ∈ ℝ*𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
 
Theoremiccdisj 45762 If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
 
20.42.7  Topology

Additional contents for topology.

 
20.42.7.1  Closure and interior
 
Theoremclduni 45763 The union of closed sets is the underlying set of the topology (the union of open sets). (Contributed by Zhi Wang, 6-Sep-2024.)
(𝐽 ∈ Top → (Clsd‘𝐽) = 𝐽)
 
Theoremopncldeqv 45764* Conditions on open sets are equivalent to conditions on closed sets. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑𝐽 ∈ Top)    &   ((𝜑𝑥 = ( 𝐽𝑦)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
 
Theoremopndisj 45765 Two ways of saying that two open sets are disjoint, if 𝐽 is a topology and 𝑋 is an open set. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝑍 = ( 𝐽𝑋) → (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌𝐽 ∧ (𝑋𝑌) = ∅)))
 
Theoremclddisj 45766 Two ways of saying that two closed sets are disjoint, if 𝐽 is a topology and 𝑋 is a closed set. An alternative proof is similar to that of opndisj 45765 with elssuni 4838 replaced by the combination of cldss 21792 and eqid 2739. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝑍 = ( 𝐽𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋𝑌) = ∅)))
 
20.42.7.2  Neighborhoods
 
Theoremneircl 45767 Reverse closure of the neighborhood operation. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by Zhi Wang, 16-Sep-2024.)
(𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top)
 
Theoremopnneilem 45768* Lemma factoring out common proof steps of opnneil 45772 and opnneirv 45770. (Contributed by Zhi Wang, 31-Aug-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐽 (𝑆𝑥𝜓) ↔ ∃𝑦𝐽 (𝑆𝑦𝜒)))
 
Theoremopnneir 45769* If something is true for an open neighborhood, it must be true for a neighborhood. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑𝐽 ∈ Top)       (𝜑 → (∃𝑥𝐽 (𝑆𝑥𝜓) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓))
 
Theoremopnneirv 45770* A variant of opnneir 45769 with different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑𝐽 ∈ Top)    &   ((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐽 (𝑆𝑥𝜓) → ∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝜒))
 
Theoremopnneilv 45771* The converse of opnneir 45769 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 45767), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑𝐽 ∈ Top)    &   ((𝜑𝑦𝑥) → (𝜓𝜒))       (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦𝐽 (𝑆𝑦𝜒)))
 
Theoremopnneil 45772* A variant of opnneilv 45771. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑𝐽 ∈ Top)    &   ((𝜑𝑦𝑥) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑥𝐽 (𝑆𝑥𝜓)))
 
Theoremopnneieqv 45773* The equivalence between neighborhood and open neighborhood. See opnneieqvv 45774 for different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑𝐽 ∈ Top)    &   ((𝜑𝑦𝑥) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥𝐽 (𝑆𝑥𝜓)))
 
Theoremopnneieqvv 45774* The equivalence between neighborhood and open neighborhood. A variant of opnneieqv 45773 with two dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑𝐽 ∈ Top)    &   ((𝜑𝑦𝑥) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑦𝐽 (𝑆𝑦𝜒)))
 
20.42.7.3  Subspace topologies
 
Theoremrestcls2lem 45775 A closed set in a subspace topology is a subset of the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑𝑋 = 𝐽)    &   (𝜑𝑌𝑋)    &   (𝜑𝐾 = (𝐽t 𝑌))    &   (𝜑𝑆 ∈ (Clsd‘𝐾))       (𝜑𝑆𝑌)
 
Theoremrestcls2 45776 A closed set in a subspace topology is the closure in the original topology intersecting with the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑𝑋 = 𝐽)    &   (𝜑𝑌𝑋)    &   (𝜑𝐾 = (𝐽t 𝑌))    &   (𝜑𝑆 ∈ (Clsd‘𝐾))       (𝜑𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
 
Theoremrestclsseplem 45777 Lemma for restclssep 45778. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑𝑋 = 𝐽)    &   (𝜑𝑌𝑋)    &   (𝜑𝐾 = (𝐽t 𝑌))    &   (𝜑𝑆 ∈ (Clsd‘𝐾))    &   (𝜑 → (𝑆𝑇) = ∅)    &   (𝜑𝑇𝑌)       (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
 
Theoremrestclssep 45778 Two disjoint closed sets in a subspace topology are separated in the original topology. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑𝑋 = 𝐽)    &   (𝜑𝑌𝑋)    &   (𝜑𝐾 = (𝐽t 𝑌))    &   (𝜑𝑆 ∈ (Clsd‘𝐾))    &   (𝜑 → (𝑆𝑇) = ∅)    &   (𝜑𝑇 ∈ (Clsd‘𝐾))       (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
 
20.42.7.4  Limits and continuity in topological spaces
 
Theoremcnneiima 45779 Given a continuous function, the preimage of a neighborhood is a neighborhood. To be precise, the preimage of a neighborhood of a subset 𝑇 of the codomain of a continuous function is a neighborhood of any subset of the preimage of 𝑇. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑁 ∈ ((nei‘𝐾)‘𝑇))    &   (𝜑𝑆 ⊆ (𝐹𝑇))       (𝜑 → (𝐹𝑁) ∈ ((nei‘𝐽)‘𝑆))
 
20.42.7.5  Topological definitions using the reals
 
Theoremiooii 45780 Open intervals are open sets of II. (Contributed by Zhi Wang, 9-Sep-2024.)
((0 ≤ 𝐴𝐵 ≤ 1) → (𝐴(,)𝐵) ∈ II)
 
Theoremicccldii 45781 Closed intervals are closed sets of II. Note that iccss 12901, iccordt 21977, and ordtresticc 21986 are proved from ixxss12 12853, ordtcld3 21962, and ordtrest2 21967, respectively. An alternate proof uses restcldi 21936, dfii2 23646, and icccld 23531. (Contributed by Zhi Wang, 8-Sep-2024.)
((0 ≤ 𝐴𝐵 ≤ 1) → (𝐴[,]𝐵) ∈ (Clsd‘II))
 
Theoremi0oii 45782 (0[,)𝐴) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝐴 ≤ 1 → (0[,)𝐴) ∈ II)
 
Theoremio1ii 45783 (𝐴(,]1) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
(0 ≤ 𝐴 → (𝐴(,]1) ∈ II)
 
20.42.7.6  Separated sets
 
Theoremsepnsepolem1 45784* Lemma for sepnsepo 45786. (Contributed by Zhi Wang, 1-Sep-2024.)
(∃𝑥𝐽𝑦𝐽 (𝜑𝜓𝜒) ↔ ∃𝑥𝐽 (𝜑 ∧ ∃𝑦𝐽 (𝜓𝜒)))
 
Theoremsepnsepolem2 45785* Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 45786. Proof could be shortened by 1 step using ssdisjdr 45733. (Contributed by Zhi Wang, 1-Sep-2024.)
(𝜑𝐽 ∈ Top)       (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
 
Theoremsepnsepo 45786* Open neighborhood and neighborhood is equivalent regarding disjointness for both sides. Namely, separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
(𝜑𝐽 ∈ Top)       (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
 
Theoremsepdisj 45787 Separated sets are disjoint. Note that in general separatedness also requires 𝑇 𝐽 and (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ as well but they are unnecessary here. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑𝑆 𝐽)    &   (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)       (𝜑 → (𝑆𝑇) = ∅)
 
Theoremseposep 45788* If two sets are separated by (open) neighborhoods, then they are separated subsets of the underlying set. Note that separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. See sepnsepo 45786. The relationship between separatedness and closure is also seen in isnrm 22098, isnrm2 22121, isnrm3 22122. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑 → ∃𝑛𝐽𝑚𝐽 (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅))       (𝜑 → ((𝑆 𝐽𝑇 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
 
Theoremsepcsepo 45789* If two sets are separated by closed neighborhoods, then they are separated by (open) neighborhoods. See sepnsepo 45786 for the equivalence between separatedness by open neighborhoods and separatedness by neighborhoods. Although 𝐽 ∈ Top might be redundant here, it is listed for explicitness. 𝐽 ∈ Top can be obtained from neircl 45767, adantr 484, and rexlimiva 3192. (Contributed by Zhi Wang, 8-Sep-2024.)
(𝜑𝐽 ∈ Top)    &   (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛𝑚) = ∅))       (𝜑 → ∃𝑛𝐽𝑚𝐽 (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅))
 
Theoremsepfsepc 45790* If two sets are separated by a continuous function, then they are separated by closed neighborhoods. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (𝑓 “ {0}) ∧ 𝑇 ⊆ (𝑓 “ {1})))       (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛𝑚) = ∅))
 
Theoremseppsepf 45791 If two sets are precisely separated by a continuous function, then they are separated by the continuous function. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (𝑓 “ {0}) ∧ 𝑇 = (𝑓 “ {1})))       (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (𝑓 “ {0}) ∧ 𝑇 ⊆ (𝑓 “ {1})))
 
Theoremseppcld 45792* If two sets are precisely separated by a continuous function, then they are closed. An alternate proof involves II ∈ Fre. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (𝑓 “ {0}) ∧ 𝑇 = (𝑓 “ {1})))       (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))
 
20.42.7.7  Separated spaces: T0, T1, T2 (Hausdorff) ...
 
Theoremisnrm4 45793* A topological space is normal iff any two disjoint closed sets are separated by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
(𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥𝑦) = ∅)))
 
Theoremdfnrm2 45794* A topological space is normal if any disjoint closed sets can be separated by open neighborhoods. An alternate definition of df-nrm 22080. (Contributed by Zhi Wang, 30-Aug-2024.)
Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐𝑑) = ∅ → ∃𝑥𝑗𝑦𝑗 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))}
 
Theoremdfnrm3 45795* A topological space is normal if any disjoint closed sets can be separated by neighborhoods. An alternate definition of df-nrm 22080. (Contributed by Zhi Wang, 2-Sep-2024.)
Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝑗)‘𝑐)∃𝑦 ∈ ((nei‘𝑗)‘𝑑)(𝑥𝑦) = ∅)}
 
Theoremiscnrm3lem1 45796* Lemma for iscnrm3 45815. Subspace topology is a topology. (Contributed by Zhi Wang, 3-Sep-2024.)
(𝐽 ∈ Top → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 ((𝐽t 𝑥) ∈ Top ∧ 𝜑)))
 
Theoremiscnrm3lem2 45797* Lemma for iscnrm3 45815 proving a biconditional on restricted universal quantifications. (Contributed by Zhi Wang, 3-Sep-2024.)
(𝜑 → (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜓 → ((𝑤𝐷𝑣𝐸) → 𝜒)))    &   (𝜑 → (∀𝑤𝐷𝑣𝐸 𝜒 → ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜓)))       (𝜑 → (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜓 ↔ ∀𝑤𝐷𝑣𝐸 𝜒))
 
Theoremiscnrm3lem3 45798 Lemma for iscnrm3lem4 45799. (Contributed by Zhi Wang, 4-Sep-2024.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒𝜃) ∧ 𝜓))
 
Theoremiscnrm3lem4 45799 Lemma for iscnrm3lem5 45800 and iscnrm3r 45811. (Contributed by Zhi Wang, 4-Sep-2024.)
(𝜂 → (𝜓𝜁))    &   ((𝜑𝜒𝜃) → 𝜂)    &   ((𝜑𝜒𝜃) → (𝜁𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremiscnrm3lem5 45800* Lemma for iscnrm3l 45814. (Contributed by Zhi Wang, 3-Sep-2024.)
((𝑥 = 𝑆𝑦 = 𝑇) → (𝜑𝜓))    &   ((𝑥 = 𝑆𝑦 = 𝑇) → (𝜒𝜃))    &   ((𝜏𝜂𝜁) → (𝑆𝑉𝑇𝑊))    &   ((𝜏𝜂𝜁) → ((𝜓𝜃) → 𝜎))       (𝜏 → (∀𝑥𝑉𝑦𝑊 (𝜑𝜒) → (𝜂 → (𝜁𝜎))))
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