HomeHome Metamath Proof Explorer
Theorem List (p. 133 of 429)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27903)
  Hilbert Space Explorer  Hilbert Space Explorer
(27904-29428)
  Users' Mathboxes  Users' Mathboxes
(29429-42879)
 

Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhashrabsn1 13201* If the size of a restricted class abstraction restricted to a singleton is 1, the condition of the class abstraction must hold for the singleton. (Contributed by Alexander van der Vekens, 3-Sep-2018.)
((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 1 → [𝐴 / 𝑥]𝜑)
 
Theoremhashfn 13202 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝐹 Fn 𝐴 → (#‘𝐹) = (#‘𝐴))
 
Theoremfseq1hash 13203 The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
((𝑁 ∈ ℕ0𝐹 Fn (1...𝑁)) → (#‘𝐹) = 𝑁)
 
Theoremhashgadd 13204 𝐺 maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)       ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +𝑜 𝐵)) = ((𝐺𝐴) + (𝐺𝐵)))
 
Theoremhashgval2 13205 A short expression for the 𝐺 function of hashgf1o 12810. (Contributed by Mario Carneiro, 24-Jan-2015.)
(# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
 
Theoremhashdom 13206 Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)
((𝐴 ∈ Fin ∧ 𝐵𝑉) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴𝐵))
 
Theoremhashdomi 13207 Non-strict order relation of the # function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝐴𝐵 → (#‘𝐴) ≤ (#‘𝐵))
 
Theoremhashsdom 13208 Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) < (#‘𝐵) ↔ 𝐴𝐵))
 
Theoremhashun 13209 The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (#‘(𝐴𝐵)) = ((#‘𝐴) + (#‘𝐵)))
 
Theoremhashun2 13210 The size of the union of finite sets is less than or equal to the sum of their sizes. (Contributed by Mario Carneiro, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 27-Jul-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴𝐵)) ≤ ((#‘𝐴) + (#‘𝐵)))
 
Theoremhashun3 13211 The size of the union of finite sets is the sum of their sizes minus the size of the intersection. (Contributed by Mario Carneiro, 6-Aug-2017.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴𝐵)) = (((#‘𝐴) + (#‘𝐵)) − (#‘(𝐴𝐵))))
 
Theoremhashinfxadd 13212 The extended real addition of the size of an infinite set with the size of an arbitrary set yields plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
((𝐴𝑉𝐵𝑊 ∧ (#‘𝐴) ∉ ℕ0) → ((#‘𝐴) +𝑒 (#‘𝐵)) = +∞)
 
Theoremhashunx 13213 The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 13209. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
((𝐴𝑉𝐵𝑊 ∧ (𝐴𝐵) = ∅) → (#‘(𝐴𝐵)) = ((#‘𝐴) +𝑒 (#‘𝐵)))
 
Theoremhashge0 13214 The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(𝐴𝑉 → 0 ≤ (#‘𝐴))
 
Theoremhashgt0 13215 The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 0 < (#‘𝐴))
 
Theoremhashge1 13216 The cardinality of a nonempty set is greater or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 1 ≤ (#‘𝐴))
 
Theorem1elfz0hash 13217 1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)
((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 1 ∈ (0...(#‘𝐴)))
 
Theoremhashnn0n0nn 13218 If a nonnegative integer is the size of a set which contains at least one element, this integer is a positive integer. (Contributed by Alexander van der Vekens, 9-Jan-2018.)
(((𝑉𝑊𝑌 ∈ ℕ0) ∧ ((#‘𝑉) = 𝑌𝑁𝑉)) → 𝑌 ∈ ℕ)
 
Theoremhashunsng 13219 The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)
(𝐵𝑉 → ((𝐴 ∈ Fin ∧ ¬ 𝐵𝐴) → (#‘(𝐴 ∪ {𝐵})) = ((#‘𝐴) + 1)))
 
Theoremhashprg 13220 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ (#‘{𝐴, 𝐵}) = 2))
 
TheoremhashprgOLD 13221 Obsolete version of hashprg 13220 as of 18-Sep-2021. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝑉𝐵𝑉) → (𝐴𝐵 ↔ (#‘{𝐴, 𝐵}) = 2))
 
Theoremelprchashprn2 13222 If one element of an unordered pair is not a set, the size of the unordered pair is not 2. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
𝑀 ∈ V → ¬ (#‘{𝑀, 𝑁}) = 2)
 
Theoremhashprb 13223 The size of an unordered pair is 2 if and only if its elements are different sets. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
((𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀𝑁) ↔ (#‘{𝑀, 𝑁}) = 2)
 
Theoremhashprdifel 13224 The elements of an unordered pair of size 2 are different sets. (Contributed by AV, 27-Jan-2020.)
𝑆 = {𝐴, 𝐵}       ((#‘𝑆) = 2 → (𝐴𝑆𝐵𝑆𝐴𝐵))
 
Theoremprhash2ex 13225 There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 13234, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)
(#‘{0, 1}) = 2
 
Theoremhashle00 13226 If the size of a set is less than or equal to zero, the set must be empty. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Proof shortened by AV, 24-Oct-2021.)
(𝑉𝑊 → ((#‘𝑉) ≤ 0 ↔ 𝑉 = ∅))
 
Theoremhashgt0elex 13227* If the size of a set is greater than zero, the set must contain at least one element. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝑉𝑊 ∧ 0 < (#‘𝑉)) → ∃𝑥 𝑥𝑉)
 
Theoremhashgt0elexb 13228* The size of a set is greater than zero if and only if the set contains at least one element. (Contributed by Alexander van der Vekens, 18-Jan-2018.)
(𝑉𝑊 → (0 < (#‘𝑉) ↔ ∃𝑥 𝑥𝑉))
 
Theoremhashp1i 13229 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
𝐴 ∈ ω    &   𝐵 = suc 𝐴    &   (#‘𝐴) = 𝑀    &   (𝑀 + 1) = 𝑁       (#‘𝐵) = 𝑁
 
Theoremhash1 13230 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘1𝑜) = 1
 
Theoremhash2 13231 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘2𝑜) = 2
 
Theoremhash3 13232 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘3𝑜) = 3
 
Theoremhash4 13233 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘4𝑜) = 4
 
Theorempr0hash2ex 13234 There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)
(#‘{∅, {∅}}) = 2
 
Theoremhashss 13235 The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
((𝐴𝑉𝐵𝐴) → (#‘𝐵) ≤ (#‘𝐴))
 
Theoremprsshashgt1 13236 The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)
(((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ 𝐶𝑈) → ({𝐴, 𝐵} ⊆ 𝐶 → 2 ≤ (#‘𝐶)))
 
Theoremhashin 13237 The size of the intersection of a set and a class is less than or equal to the size of the set. (Contributed by AV, 4-Jan-2021.)
(𝐴𝑉 → (#‘(𝐴𝐵)) ≤ (#‘𝐴))
 
Theoremhashssdif 13238 The size of the difference of a finite set and a subset is the set's size minus the subset's. (Contributed by Steve Rodriguez, 24-Oct-2015.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (#‘(𝐴𝐵)) = ((#‘𝐴) − (#‘𝐵)))
 
Theoremhashdif 13239 The size of the difference of a finite set and another set is the first set's size minus that of the intersection of both. (Contributed by Steve Rodriguez, 24-Oct-2015.)
(𝐴 ∈ Fin → (#‘(𝐴𝐵)) = ((#‘𝐴) − (#‘(𝐴𝐵))))
 
Theoremhashdifsn 13240 The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (#‘(𝐴 ∖ {𝐵})) = ((#‘𝐴) − 1))
 
Theoremhashdifpr 13241 The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)
((𝐴 ∈ Fin ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → (#‘(𝐴 ∖ {𝐵, 𝐶})) = ((#‘𝐴) − 2))
 
Theoremhashsn01 13242 The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.)
((#‘{𝐴}) = 0 ∨ (#‘{𝐴}) = 1)
 
Theoremhashsnle1 13243 The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.)
(#‘{𝐴}) ≤ 1
 
Theoremhashsnlei 13244 Get an upper bound on a concretely specified finite set. Base case: singleton set. (Contributed by Mario Carneiro, 11-Feb-2015.) (Proof shortened by AV, 23-Feb-2021.)
({𝐴} ∈ Fin ∧ (#‘{𝐴}) ≤ 1)
 
Theoremhash1snb 13245* The size of a set is 1 if and only if it is a singleton (containing a set). (Contributed by Alexander van der Vekens, 7-Dec-2017.)
(𝑉𝑊 → ((#‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎}))
 
Theoremeuhash1 13246* The size of a set is 1 in terms of existential uniqueness. (Contributed by Alexander van der Vekens, 8-Feb-2018.)
(𝑉𝑊 → ((#‘𝑉) = 1 ↔ ∃!𝑎 𝑎𝑉))
 
Theoremhash1n0 13247 If the size of a set is 1 the set is not empty. (Contributed by AV, 23-Dec-2020.)
((𝐴𝑉 ∧ (#‘𝐴) = 1) → 𝐴 ≠ ∅)
 
Theoremhashgt12el 13248* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
((𝑉𝑊 ∧ 1 < (#‘𝑉)) → ∃𝑎𝑉𝑏𝑉 𝑎𝑏)
 
Theoremhashgt12el2 13249* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
((𝑉𝑊 ∧ 1 < (#‘𝑉) ∧ 𝐴𝑉) → ∃𝑏𝑉 𝐴𝑏)
 
Theoremhashunlei 13250 Get an upper bound on a concretely specified finite set. Induction step: union of two finite bounded sets. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐶 = (𝐴𝐵)    &   (𝐴 ∈ Fin ∧ (#‘𝐴) ≤ 𝐾)    &   (𝐵 ∈ Fin ∧ (#‘𝐵) ≤ 𝑀)    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝐾 + 𝑀) = 𝑁       (𝐶 ∈ Fin ∧ (#‘𝐶) ≤ 𝑁)
 
Theoremhashsslei 13251 Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐵𝐴    &   (𝐴 ∈ Fin ∧ (#‘𝐴) ≤ 𝑁)    &   𝑁 ∈ ℕ0       (𝐵 ∈ Fin ∧ (#‘𝐵) ≤ 𝑁)
 
Theoremhashfz 13252 Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)
(𝐵 ∈ (ℤ𝐴) → (#‘(𝐴...𝐵)) = ((𝐵𝐴) + 1))
 
Theoremfzsdom2 13253 Condition for finite ranges to have a strict dominance relation. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2015.)
(((𝐵 ∈ (ℤ𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶))
 
Theoremhashfzo 13254 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (#‘(𝐴..^𝐵)) = (𝐵𝐴))
 
Theoremhashfzo0 13255 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐵 ∈ ℕ0 → (#‘(0..^𝐵)) = 𝐵)
 
Theoremhashfzp1 13256 Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
(𝐵 ∈ (ℤ𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵𝐴))
 
Theoremhashfz0 13257 Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝐵 ∈ ℕ0 → (#‘(0...𝐵)) = (𝐵 + 1))
 
Theoremhashxplem 13258 Lemma for hashxp 13259. (Contributed by Paul Chapman, 30-Nov-2012.)
𝐵 ∈ Fin       (𝐴 ∈ Fin → (#‘(𝐴 × 𝐵)) = ((#‘𝐴) · (#‘𝐵)))
 
Theoremhashxp 13259 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 × 𝐵)) = ((#‘𝐴) · (#‘𝐵)))
 
Theoremhashmap 13260 The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴𝑚 𝐵)) = ((#‘𝐴)↑(#‘𝐵)))
 
Theoremhashpw 13261 The size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) (Proof shortened by Mario Carneiro, 5-Aug-2014.)
(𝐴 ∈ Fin → (#‘𝒫 𝐴) = (2↑(#‘𝐴)))
 
Theoremhashfun 13262 A finite set is a function iff it is equinumerous to its domain. (Contributed by Mario Carneiro, 26-Sep-2013.) (Revised by Mario Carneiro, 12-Mar-2015.)
(𝐹 ∈ Fin → (Fun 𝐹 ↔ (#‘𝐹) = (#‘dom 𝐹)))
 
Theoremhashres 13263 The number of elements of a finite function restricted to a subset of its domain is equal to the number of elements of that subset. (Contributed by AV, 15-Dec-2021.)
((Fun 𝐴𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (#‘(𝐴𝐵)) = (#‘𝐵))
 
Theoremhashreshashfun 13264 The number of elements of a finite function expressed by a restriction. (Contributed by AV, 15-Dec-2021.)
((Fun 𝐴𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (#‘𝐴) = ((#‘(𝐴𝐵)) + (#‘(dom 𝐴𝐵))))
 
Theoremhashimarn 13265 The size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 equals the size of the function 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) (Proof shortened by AV, 4-May-2021.)
((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → (#‘(𝐸 “ ran 𝐹)) = (#‘𝐹)))
 
Theoremhashimarni 13266 If the size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 is a nonnegative integer, the size of the function 𝐹 is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃 = (𝐸 “ ran 𝐹) ∧ (#‘𝑃) = 𝑁) → (#‘𝐹) = 𝑁))
 
Theoremresunimafz0 13267 TODO-AV: Revise using 𝐹 ∈ Word dom 𝐼? Formerly part of proof of eupth2lem3 27214: The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
(𝜑 → Fun 𝐼)    &   (𝜑𝐹:(0..^(#‘𝐹))⟶dom 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))       (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
 
Theoremfnfz0hash 13268 The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0...𝑁)) → (#‘𝐹) = (𝑁 + 1))
 
Theoremffz0hash 13269 The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)
((𝑁 ∈ ℕ0𝐹:(0...𝑁)⟶𝐵) → (#‘𝐹) = (𝑁 + 1))
 
Theoremfnfz0hashnn0 13270 The size of a function on a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.)
(𝐹 Fn (0...𝑁) → (#‘𝐹) ∈ ℕ0)
 
Theoremffzo0hash 13271 The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0..^𝑁)) → (#‘𝐹) = 𝑁)
 
Theoremfnfzo0hash 13272 The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)
((𝑁 ∈ ℕ0𝐹:(0..^𝑁)⟶𝐵) → (#‘𝐹) = 𝑁)
 
Theoremfnfzo0hashnn0 13273 The value of the size function on a half-open range of nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.)
(𝐹 Fn (0..^𝑁) → (#‘𝐹) ∈ ℕ0)
 
Theoremhashbclem 13274* Lemma for hashbc 13275: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ∀𝑗 ∈ ℤ ((#‘𝐴)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗}))    &   (𝜑𝐾 ∈ ℤ)       (𝜑 → ((#‘(𝐴 ∪ {𝑧}))C𝐾) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (#‘𝑥) = 𝐾}))
 
Theoremhashbc 13275* The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). (Contributed by Mario Carneiro, 13-Jul-2014.)
((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
 
Theoremhashfacen 13276* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
((𝐴𝐵𝐶𝐷) → {𝑓𝑓:𝐴1-1-onto𝐶} ≈ {𝑓𝑓:𝐵1-1-onto𝐷})
 
Theoremhashf1lem1 13277* Lemma for hashf1 13279. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵))    &   (𝜑𝐹:𝐴1-1𝐵)       (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (𝐵 ∖ ran 𝐹))
 
Theoremhashf1lem2 13278* Lemma for hashf1 13279. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵))       (𝜑 → (#‘{𝑓𝑓:(𝐴 ∪ {𝑧})–1-1𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓𝑓:𝐴1-1𝐵})))
 
Theoremhashf1 13279* The permutation number 𝐴 ∣ ! · ( ∣ 𝐵 ∣ C ∣ 𝐴 ∣ ) = 𝐵 ∣ ! / ( ∣ 𝐵 ∣ − ∣ 𝐴 ∣ )! counts the number of injections from 𝐴 to 𝐵. (Contributed by Mario Carneiro, 21-Jan-2015.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘{𝑓𝑓:𝐴1-1𝐵}) = ((!‘(#‘𝐴)) · ((#‘𝐵)C(#‘𝐴))))
 
Theoremhashfac 13280* A factorial counts the number of bijections on a finite set. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by Mario Carneiro, 17-Apr-2015.)
(𝐴 ∈ Fin → (#‘{𝑓𝑓:𝐴1-1-onto𝐴}) = (!‘(#‘𝐴)))
 
Theoremleiso 13281 Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
 
Theoremleisorel 13282 Version of isorel 6616 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))
 
Theoremfz1isolem 13283* Lemma for fz1iso 13284. (Contributed by Mario Carneiro, 2-Apr-2014.)
𝐺 = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)    &   𝐵 = (ℕ ∩ ( < “ {((#‘𝐴) + 1)}))    &   𝐶 = (ω ∩ (𝐺‘((#‘𝐴) + 1)))    &   𝑂 = OrdIso(𝑅, 𝐴)       ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(#‘𝐴)), 𝐴))
 
Theoremfz1iso 13284* Any finite ordered set has an order isometry to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(#‘𝐴)), 𝐴))
 
Theoremishashinf 13285* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 8214. (Contributed by Thierry Arnoux, 5-Jul-2017.)
𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛)
 
Theoremseqcoll 13286* The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.)
((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))    &   (𝜑𝑁 ∈ (1...(#‘𝐴)))    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)    &   ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))       (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))
 
Theoremseqcoll2 13287* The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 13-Dec-2014.)
((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)    &   ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘𝐴)))
 
5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)
 
Theoremhashprlei 13288 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
({𝐴, 𝐵} ∈ Fin ∧ (#‘{𝐴, 𝐵}) ≤ 2)
 
Theoremhash2pr 13289* A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑉𝑊 ∧ (#‘𝑉) = 2) → ∃𝑎𝑏 𝑉 = {𝑎, 𝑏})
 
Theoremhash2prde 13290* A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑉𝑊 ∧ (#‘𝑉) = 2) → ∃𝑎𝑏(𝑎𝑏𝑉 = {𝑎, 𝑏}))
 
Theoremhash2exprb 13291* A set of size two is an unordered pair if and only if it contains two different elements. (Contributed by Alexander van der Vekens, 14-Jan-2018.)
(𝑉𝑊 → ((#‘𝑉) = 2 ↔ ∃𝑎𝑏(𝑎𝑏𝑉 = {𝑎, 𝑏})))
 
Theoremhash2prb 13292* A set of size two is a proper unordered pair. (Contributed by AV, 1-Nov-2020.)
(𝑉𝑊 → ((#‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))
 
Theoremprprrab 13293 The set of proper pairs of elements of a given set expressed in two ways. (Contributed by AV, 24-Nov-2020.)
{𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 2}
 
Theoremnehash2 13294 The cardinality of a set with two distinct elements. (Contributed by Thierry Arnoux, 27-Aug-2019.)
(𝜑𝑃𝑉)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)       (𝜑 → 2 ≤ (#‘𝑃))
 
Theoremhash2prd 13295 A set of size two is an unordered pair if it contains two different elements. (Contributed by Alexander van der Vekens, 9-Dec-2018.) (Proof shortened by AV, 1-Nov-2020.)
((𝑃𝑉 ∧ (#‘𝑃) = 2) → ((𝑋𝑃𝑌𝑃𝑋𝑌) → 𝑃 = {𝑋, 𝑌}))
 
Theoremhash2pwpr 13296 If the size of a subset of an unordered pair is 2, the subset is the pair itself. (Contributed by Alexander van der Vekens, 9-Dec-2018.)
(((#‘𝑃) = 2 ∧ 𝑃 ∈ 𝒫 {𝑋, 𝑌}) → 𝑃 = {𝑋, 𝑌})
 
Theoremhashle2pr 13297* A nonempty set of size less than or equal to two is an unordered pair of sets. (Contributed by AV, 24-Nov-2021.)
((𝑃𝑉𝑃 ≠ ∅) → ((#‘𝑃) ≤ 2 ↔ ∃𝑎𝑏 𝑃 = {𝑎, 𝑏}))
 
Theoremhashle2prv 13298* A nonempty subset of a powerset of a class 𝑉 has size less than or equal to two iff it is an unordered pair of elements of 𝑉. (Contributed by AV, 24-Nov-2021.)
(𝑃 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝑃) ≤ 2 ↔ ∃𝑎𝑉𝑏𝑉 𝑃 = {𝑎, 𝑏}))
 
Theorempr2pwpr 13299* The set of subsets of a pair having length 2 is the set of the pair as singleton. (Contributed by AV, 9-Dec-2018.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝑝 ∈ 𝒫 {𝐴, 𝐵} ∣ 𝑝 ≈ 2𝑜} = {{𝐴, 𝐵}})
 
Theoremhashge2el2dif 13300* A set with size at least 2 has at least 2 different elements. (Contributed by AV, 18-Mar-2019.)
((𝐷𝑉 ∧ 2 ≤ (#‘𝐷)) → ∃𝑥𝐷𝑦𝐷 𝑥𝑦)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
  Copyright terms: Public domain < Previous  Next >