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Theorem List for Metamath Proof Explorer - 36801-36900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisdomn3 36801 Nonzero elements form a multiplicative submonoid of any domain. (Contributed by Stefan O'Rear, 11-Sep-2015.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑈 = (mulGrp‘𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ Ring ∧ (𝐵 ∖ { 0 }) ∈ (SubMnd‘𝑈)))

Theoremmon1pid 36802 Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑃 = (Poly1𝑅)    &    1 = (1r𝑃)    &   𝑀 = (Monic1p𝑅)    &   𝐷 = ( deg1𝑅)       (𝑅 ∈ NzRing → ( 1𝑀 ∧ (𝐷1 ) = 0))

Theoremmon1psubm 36803 Monic polynomials are a multiplicative submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑃 = (Poly1𝑅)    &   𝑀 = (Monic1p𝑅)    &   𝑈 = (mulGrp‘𝑃)       (𝑅 ∈ NzRing → 𝑀 ∈ (SubMnd‘𝑈))

Theoremdeg1mhm 36804 Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝑌 = ((mulGrp‘𝑃) ↾s (𝐵 ∖ { 0 }))    &   𝑁 = (ℂflds0)       (𝑅 ∈ Domn → (𝐷 ↾ (𝐵 ∖ { 0 })) ∈ (𝑌 MndHom 𝑁))

Theoremcytpfn 36805 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CytP Fn ℕ

Theoremcytpval 36806* Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝑇 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    &   𝑂 = (od‘𝑇)    &   𝑃 = (Poly1‘ℂfld)    &   𝑋 = (var1‘ℂfld)    &   𝑄 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)       (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))

21.24.50  Miscellaneous topology

Theoremfgraphopab 36807* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})

Theoremfgraphxp 36808* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹:𝐴𝐵𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)})

Theoremhausgraph 36809 The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.)
((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (Clsd‘(𝐽 ×t 𝐾)))

Syntaxctopsep 36810 The class of separable toplogies.
class TopSep

Syntaxctoplnd 36811 The class of Lindelöf toplogies.
class TopLnd

Definitiondf-topsep 36812* A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.)
TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = 𝑗)}

Definitiondf-toplnd 36813* A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 = 𝑦 → ∃𝑧 ∈ 𝒫 𝑥(𝑧 ≼ ω ∧ 𝑥 = 𝑧))}

21.25  Mathbox for Jon Pennant

Theoremioounsn 36814 The closure of the upper end of an open real interval. (Contributed by Jon Pennant, 8-Jun-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))

Theoremiocunico 36815 Split an open interval into two pieces at point B, Co-author TA. (Contributed by Jon Pennant, 8-Jun-2019.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶))

Theoremiocinico 36816 The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝐵})

Theoremiocmbl 36817 An open-below, closed-above real interval is measurable. (Contributed by Jon Pennant, 12-Jun-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → (𝐴(,]𝐵) ∈ dom vol)

Theoremcnioobibld 36818* A bounded, continuous function on an open bounded interval is integrable. The function must be bounded. For a counterexample, consider 𝐹 = (𝑥 ∈ (0(,)1) ↦ (1 / 𝑥)). See cniccibl 23413 for closed bounded intervals. (Contributed by Jon Pennant, 31-May-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹𝑦)) ≤ 𝑥)       (𝜑𝐹 ∈ 𝐿1)

Theoremitgpowd 36819* The integral of a monomial on a closed bounded interval of the real line. Co-authors TA and MC. (Contributed by Jon Pennant, 31-May-2019.) (Revised by Thierry Arnoux, 14-Jun-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ∫(𝐴[,]𝐵)(𝑥𝑁) d𝑥 = (((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1))) / (𝑁 + 1)))

Theoremarearect 36820 The area of a rectangle whose sides are parallel to the coordinate axes in (ℝ × ℝ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐷 ∈ ℝ    &   𝐴𝐵    &   𝐶𝐷    &   𝑆 = ((𝐴[,]𝐵) × (𝐶[,]𝐷))       (area‘𝑆) = ((𝐵𝐴) · (𝐷𝐶))

Theoremareaquad 36821* The area of a quadrilateral with two sides which are parallel to the y-axis in (ℝ × ℝ) is its width multiplied by the average height of its higher edge minus the average height of its lower edge. Co-author TA. (Contributed by Jon Pennant, 31-May-2019.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐷 ∈ ℝ    &   𝐸 ∈ ℝ    &   𝐹 ∈ ℝ    &   𝐴 < 𝐵    &   𝐶𝐸    &   𝐷𝐹    &   𝑈 = (𝐶 + (((𝑥𝐴) / (𝐵𝐴)) · (𝐷𝐶)))    &   𝑉 = (𝐸 + (((𝑥𝐴) / (𝐵𝐴)) · (𝐹𝐸)))    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝑈[,]𝑉))}       (area‘𝑆) = ((((𝐹 + 𝐸) / 2) − ((𝐷 + 𝐶) / 2)) · (𝐵𝐴))

21.26  Mathbox for Richard Penner

21.26.1  Short Studies

21.26.1.1  Additional work on conditional logical operator

Theoremifpan123g 36822 Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑𝜒) ∧ (𝜑𝜏)) ∧ ((¬ 𝜓𝜃) ∧ (𝜓𝜂))))

Theoremifpan23 36823 Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓𝜃), (𝜒𝜏)))

Theoremifpdfor2 36824 Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜑, 𝜓))

Theoremifporcor 36825 Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
(if-(𝜑, 𝜑, 𝜓) ↔ if-(𝜓, 𝜓, 𝜑))

Theoremifpdfan2 36826 Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜑))

Theoremifpancor 36827 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))

Theoremifpdfor 36828 Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ⊤, 𝜓))

Theoremifpdfan 36829 Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ⊥))

Theoremifpbi2 36830 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))

Theoremifpbi3 36831 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))

Theoremifpim1 36832 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))

Theoremifpnot 36833 Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
𝜑 ↔ if-(𝜑, ⊥, ⊤))

Theoremifpid2 36834 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
(𝜑 ↔ if-(𝜑, ⊤, ⊥))

Theoremifpim2 36835 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))

Theoremifpbi23 36836 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))

Theoremifpdfbi 36837 Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

Theoremifpbiidcor 36838 Restatement of biid 250. (Contributed by RP, 25-Apr-2020.)
if-(𝜑, 𝜑, ¬ 𝜑)

Theoremifpbicor 36839 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑))

Theoremifpxorcor 36840 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑))

Theoremifpbi1 36841 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)))

Theoremifpnot23 36842 Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
(¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))

Theoremifpnotnotb 36843 Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ¬ if-(𝜑, 𝜓, 𝜒))

Theoremifpnorcor 36844 Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜑, ¬ 𝜓) ↔ if-(𝜓, ¬ 𝜓, ¬ 𝜑))

Theoremifpnancor 36845 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜑, ¬ 𝜓))

Theoremifpnot23b 36846 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒))

Theoremifpbiidcor2 36847 Restatement of biid 250. (Contributed by RP, 25-Apr-2020.)
¬ if-(𝜑, ¬ 𝜑, 𝜑)

Theoremifpnot23c 36848 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒))

Theoremifpnot23d 36849 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒))

Theoremifpdfnan 36850 Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, ⊤))

Theoremifpdfxor 36851 Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜓))

Theoremifpbi12 36852 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜏)))

Theoremifpbi13 36853 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃)))

Theoremifpbi123 36854 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))

Theoremifpidg 36855 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜃 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑𝜓) → 𝜃) ∧ ((𝜑𝜃) → 𝜓)) ∧ ((𝜒 → (𝜑𝜃)) ∧ (𝜃 → (𝜑𝜒)))))

Theoremifpid3g 36856 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑𝜓) → 𝜒) ∧ ((𝜑𝜒) → 𝜓)))

Theoremifpid2g 36857 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜓 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜓))))

Theoremifpid1g 36858 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜒𝜑) ∧ (𝜑𝜓)))

Theoremifpim23g 36859 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(((𝜑𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ (((𝜑𝜓) → 𝜒) ∧ (𝜒 → (𝜑𝜓))))

Theoremifpim3 36860 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑))

Theoremifpnim1 36861 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(¬ (𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑))

Theoremifpim4 36862 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑))

Theoremifpnim2 36863 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(¬ (𝜑𝜓) ↔ if-(𝜓, ¬ 𝜓, 𝜑))

Theoremifpim123g 36864 Implication of conditional logical operators. The right hand side is basically conjunctive normal form which is useful in proofs. (Contributed by RP, 16-Apr-2020.)
((if-(𝜑, 𝜒, 𝜏) → if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (𝜏𝜂)))))

Theoremifpim1g 36865 Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ (((𝜓𝜑) ∨ (𝜃𝜒)) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))

Theoremifp1bi 36866 Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.)
((if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)) ↔ ((((𝜑𝜓) ∨ (𝜒𝜃)) ∧ ((𝜑𝜓) ∨ (𝜃𝜒))) ∧ (((𝜓𝜑) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜃𝜒)))))

Theoremifpbi1b 36867 When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜒, 𝜒) ↔ if-(𝜓, 𝜒, 𝜒))

Theoremifpimimb 36868 Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))

Theoremifpororb 36869 Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∨ if-(𝜑, 𝜒, 𝜏)))

Theoremifpananb 36870 Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∧ if-(𝜑, 𝜒, 𝜏)))

Theoremifpnannanb 36871 Factor conditional logic operator over nand in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊼ if-(𝜑, 𝜒, 𝜏)))

Theoremifpor123g 36872 Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (𝜏𝜂)))))

Theoremifpimim 36873 Consequnce of implication. (Contributed by RP, 17-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))

Theoremifpbibib 36874 Factor conditional logic operator over biimplication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)))

Theoremifpxorxorb 36875 Factor conditional logic operator over xor in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊻ if-(𝜑, 𝜒, 𝜏)))

21.26.1.2  Sophisms

Theoremrp-fakeimass 36876 A special case where implication appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
((𝜑𝜒) ↔ (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))

Theoremrp-fakeanorass 36877 A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
((𝜒𝜑) ↔ (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒))))

Theoremrp-fakeoranass 36878 A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
((𝜑𝜒) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))

Theoremrp-fakenanass 36879 A special case where nand appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
((𝜑𝜒) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))

Theoremrp-fakeinunass 36880 A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
(𝐶𝐴 ↔ ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)))

Theoremrp-fakeuninass 36881 A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
(𝐴𝐶 ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))

21.26.1.3  Finite Sets

Membership in the class of finite sets can be expressed in many ways.

Theoremrp-isfinite5 36882* A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ0. (Contributed by Richard Penner, 3-Mar-2020.)
(𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)

Theoremrp-isfinite6 36883* A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ. (Contributed by Richard Penner, 10-Mar-2020.)
(𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))

21.26.1.4  Infinite Sets

Theorempwelg 36884* The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.)
(∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))

Theorempwinfig 36885* The powerclass of an infinite set is an infinite set, and vice-versa. Here 𝐵 is a class which is closed under both the union and the powerclass operations and which may have infinite sets as members. (Contributed by RP, 21-Mar-2020.)
(∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴 ∈ (𝐵 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝐵 ∖ Fin)))

Theorempwinfi2 36886 The powerclass of an infinite set is an infinite set, and vice-versa. Here 𝑈 is a weak universe. (Contributed by RP, 21-Mar-2020.)
(𝑈 ∈ WUni → (𝐴 ∈ (𝑈 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝑈 ∖ Fin)))

Theorempwinfi3 36887 The powerclass of an infinite set is an infinite set, and vice-versa. Here 𝑇 is a transitive Tarski universe. (Contributed by RP, 21-Mar-2020.)
((𝑇 ∈ Tarski ∧ Tr 𝑇) → (𝐴 ∈ (𝑇 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝑇 ∖ Fin)))

Theorempwinfi 36888 The powerclass of an infinite set is an infinite set, and vice-versa. (Contributed by RP, 21-Mar-2020.)
(𝐴 ∈ (V ∖ Fin) ↔ 𝒫 𝐴 ∈ (V ∖ Fin))

21.26.1.5  Finite intersection property

While there is not yet a definition, the finite intersection property of a class is introduced by fiint 8122 where two textbook definitions are shown to be equivalent.

This property is seen often with ordinal numbers (onin 5671, ordelinel 5742 ), chains of sets ordered by the proper subset relation (sorpssin 6843), various sets in the field of topology (inopn 20529, incld 20657, innei 20739, ... ) and "universal" classes like weak universes (wunin 9414, tskin 9460) and the class of all sets (inex1g 4729) .

Theoremfipjust 36889* A definition of the finite intersection property of a class based on closure under pair-wise intersection of its elements is independent of the dummy variables. (Contributed by Richard Penner, 1-Jan-2020.)
(∀𝑢𝐴𝑣𝐴 (𝑢𝑣) ∈ 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)

Theoremcllem0 36890* The class of all sets with property 𝜑(𝑧) is closed under the binary operation on sets defined in 𝑅(𝑥, 𝑦). (Contributed by Richard Penner, 3-Jan-2020.)
𝑉 = {𝑧𝜑}    &   𝑅𝑈    &   (𝑧 = 𝑅 → (𝜑𝜓))    &   (𝑧 = 𝑥 → (𝜑𝜒))    &   (𝑧 = 𝑦 → (𝜑𝜃))    &   ((𝜒𝜃) → 𝜓)       𝑥𝑉𝑦𝑉 𝑅𝑉

Theoremsuperficl 36891* The class of all supersets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
𝐴 = {𝑧𝐵𝑧}       𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴

Theoremsuperuncl 36892* The class of all supersets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.)
𝐴 = {𝑧𝐵𝑧}       𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴

Theoremssficl 36893* The class of all subsets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
𝐴 = {𝑧𝑧𝐵}       𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴

Theoremssuncl 36894* The class of all subsets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.)
𝐴 = {𝑧𝑧𝐵}       𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴

Theoremssdifcl 36895* The class of all subsets of a class is closed under class difference. (Contributed by Richard Penner, 3-Jan-2020.)
𝐴 = {𝑧𝑧𝐵}       𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴

Theoremsssymdifcl 36896* The class of all subsets of a class is closed under symmetric difference. (Contributed by Richard Penner, 3-Jan-2020.)
𝐴 = {𝑧𝑧𝐵}       𝑥𝐴𝑦𝐴 ((𝑥𝑦) ∪ (𝑦𝑥)) ∈ 𝐴

Theoremfiinfi 36897* If two classes have the finite intersection property, then so does their intersection. (Contributed by Richard Penner, 1-Jan-2020.)
(𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)    &   (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵)    &   (𝜑𝐶 = (𝐴𝐵))       (𝜑 → ∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶)

21.26.1.6  RP ADDTO: Subclasses and subsets

Theoremrababg 36898 Condition when restricted class is equal to unrestricted class. (Contributed by RP, 13-Aug-2020.)
(∀𝑥(𝜑𝑥𝐴) ↔ {𝑥𝐴𝜑} = {𝑥𝜑})

21.26.1.7  RP ADDTO: The intersection of a class

Theoremelintabg 36899* Two ways of saying a set is an element of the intersection of a class. (Contributed by RP, 13-Aug-2020.)
(𝐴𝑉 → (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥)))

Theoremelinintab 36900* Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020.)
(𝐴 ∈ (𝐵 {𝑥𝜑}) ↔ (𝐴𝐵 ∧ ∀𝑥(𝜑𝐴𝑥)))

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