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Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembcn2 13101 Binomial coefficient: 𝑁 choose 2. (Contributed by Mario Carneiro, 22-May-2014.)
(𝑁 ∈ ℕ0 → (𝑁C2) = ((𝑁 · (𝑁 − 1)) / 2))

Theorembcp1m1 13102 Compute the binomial coefficient of (𝑁 + 1) over (𝑁 − 1) (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)
(𝑁 ∈ ℕ0 → ((𝑁 + 1)C(𝑁 − 1)) = (((𝑁 + 1) · 𝑁) / 2))

Theorembcpasc 13103 Pascal's rule for the binomial coefficient, generalized to all integers 𝐾. Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾))

Theorembccl 13104 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ0)

Theorembccl2 13105 A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)
(𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℕ)

Theorembcn2m1 13106 Compute the binomial coefficient "𝑁 choose 2 " from "(𝑁 − 1) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.)
(𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (𝑁C2))

Theorembcn2p1 13107 Compute the binomial coefficient "(𝑁 + 1) choose 2 " from "𝑁 choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der Vekens, 8-Jan-2018.)
(𝑁 ∈ ℕ0 → (𝑁 + (𝑁C2)) = ((𝑁 + 1)C2))

Theorempermnn 13108 The number of permutations of 𝑁𝑅 objects from a collection of 𝑁 objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.)
(𝑅 ∈ (0...𝑁) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ)

Theorembcnm1 13109 The binomial coefficent of (𝑁 − 1) is 𝑁. (Contributed by Scott Fenton, 16-May-2014.)
(𝑁 ∈ ℕ0 → (𝑁C(𝑁 − 1)) = 𝑁)

Theorem4bc3eq4 13110 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
(4C3) = 4

Theorem4bc2eq6 13111 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
(4C2) = 6

5.6.11  The ` # ` (set size) function

Syntaxchash 13112 Extend the definition of a class to include the set size function.
class #

Definitiondf-hash 13113 Define the set size function #, which gives the cardinality of a finite set as a member of 0, and assigns all infinite sets the value +∞. For example, (#‘{0, 1, 2}) = 3 (ex-hash 27294). (Contributed by Paul Chapman, 22-Jun-2011.)
# = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))

Theoremhashkf 13114 The finite part of the size function maps all finite sets to their cardinality, as members of 0. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)    &   𝐾 = (𝐺 ∘ card)       𝐾:Fin⟶ℕ0

Theoremhashgval 13115* The value of the # function in terms of the mapping 𝐺 from ω to 0. The proof avoids the use of ax-ac 9278. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 26-Dec-2014.)
𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)       (𝐴 ∈ Fin → (𝐺‘(card‘𝐴)) = (#‘𝐴))

Theoremhashginv 13116* 𝐺 maps the size function's value to card. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)       (𝐴 ∈ Fin → (𝐺‘(#‘𝐴)) = (card‘𝐴))

Theoremhashinf 13117 The value of the # function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.)
((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞)

Theoremhashbnd 13118 If 𝐴 has size bounded by an integer 𝐵, then 𝐴 is finite. (Contributed by Mario Carneiro, 14-Jun-2015.)
((𝐴𝑉𝐵 ∈ ℕ0 ∧ (#‘𝐴) ≤ 𝐵) → 𝐴 ∈ Fin)

Theoremhashfxnn0 13119 The size function is a function into the extended nonnegative integers. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by AV, 10-Dec-2020.)
#:V⟶ℕ0*

Theoremhashf 13120 The size function maps all finite sets to their cardinality, as members of 0, and infinite sets to +∞. TODO-AV: mark as OBSOLETE and replace it by hashfxnn0 13119? (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.) (Proof shortened by AV, 24-Oct-2021.)
#:V⟶(ℕ0 ∪ {+∞})

TheoremhashfOLD 13121 Obsolete version of hashf 13120 as of 24-Oct-2021. The size function maps all finite sets to their cardinality, as members of 0, and infinite sets to +∞. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
#:V⟶(ℕ0 ∪ {+∞})

Theoremhashxnn0 13122 The value of the hash function for a set is an extended nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Dec-2020.)
(𝑀𝑉 → (#‘𝑀) ∈ ℕ0*)

Theoremhashresfn 13123 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(# ↾ 𝐴) Fn 𝐴

Theoremdmhashres 13124 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017.)
dom (# ↾ 𝐴) = 𝐴

Theoremhashnn0pnf 13125 The value of the hash function for a set is either a nonnegative integer or positive infinity. TODO-AV: mark as OBSOLETE and replace it by hashxnn0 13122? (Contributed by Alexander van der Vekens, 6-Dec-2017.)
(𝑀𝑉 → ((#‘𝑀) ∈ ℕ0 ∨ (#‘𝑀) = +∞))

Theoremhashnnn0genn0 13126 If the size of a set is not a nonnegative integer, it is greater than or equal to any nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
((𝑀𝑉 ∧ (#‘𝑀) ∉ ℕ0𝑁 ∈ ℕ0) → 𝑁 ≤ (#‘𝑀))

Theoremhashnemnf 13127 The size of a set is never minus infinity. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
(𝐴𝑉 → (#‘𝐴) ≠ -∞)

Theoremhashv01gt1 13128 The size of a set is either 0 or 1 or greater than 1. (Contributed by Alexander van der Vekens, 29-Dec-2017.)
(𝑀𝑉 → ((#‘𝑀) = 0 ∨ (#‘𝑀) = 1 ∨ 1 < (#‘𝑀)))

Theoremhashfz1 13129 The set (1...𝑁) has 𝑁 elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
(𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)

Theoremhashen 13130 Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) = (#‘𝐵) ↔ 𝐴𝐵))

Theoremhasheni 13131 Equinumerous sets have the same number of elements (even if they are not finite). (Contributed by Mario Carneiro, 15-Apr-2015.)
(𝐴𝐵 → (#‘𝐴) = (#‘𝐵))

Theoremhasheqf1o 13132* The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) = (#‘𝐵) ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))

Theoremfiinfnf1o 13133* There is no bijection between a finite set and an infinite set. (Contributed by Alexander van der Vekens, 25-Dec-2017.)
((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴1-1-onto𝐵)

Theoremfocdmex 13134 The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.)
((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)

Theoremhasheqf1oi 13135* The size of two sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 25-Dec-2017.) (Revised by AV, 4-May-2021.)
(𝐴𝑉 → (∃𝑓 𝑓:𝐴1-1-onto𝐵 → (#‘𝐴) = (#‘𝐵)))

Theoremhasheqf1oiOLD 13136* Obsolete version of hasheqf1oi 13135 as of 4-May-2021. (Contributed by Alexander van der Vekens, 25-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓:𝐴1-1-onto𝐵 → (#‘𝐴) = (#‘𝐵)))

Theoremhashf1rn 13137 The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 4-May-2021.)
((𝐴𝑉𝐹:𝐴1-1𝐵) → (#‘𝐹) = (#‘ran 𝐹))

Theoremhashf1rnOLD 13138 Obsolete version of hashf1rn 13137 as of 4-May-2021. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by Alexander van der Vekens, 4-Feb-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐹:𝐴1-1𝐵 → (#‘𝐹) = (#‘ran 𝐹)))

Theoremhasheqf1od 13139 The size of two sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by AV, 4-May-2021.)
(𝜑𝐴𝑈)    &   (𝜑𝐹:𝐴1-1-onto𝐵)       (𝜑 → (#‘𝐴) = (#‘𝐵))

Theoremfz1eqb 13140 Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((1...𝑀) = (1...𝑁) ↔ 𝑀 = 𝑁))

Theoremhashcard 13141 The size function of the cardinality function. (Contributed by Mario Carneiro, 19-Sep-2013.) (Revised by Mario Carneiro, 4-Nov-2013.)
(𝐴 ∈ Fin → (#‘(card‘𝐴)) = (#‘𝐴))

Theoremhashcl 13142 Closure of the # function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
(𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)

Theoremhashxrcl 13143 Extended real closure of the # function. (Contributed by Mario Carneiro, 22-Apr-2015.)
(𝐴𝑉 → (#‘𝐴) ∈ ℝ*)

Theoremhashclb 13144 Reverse closure of the # function. (Contributed by Mario Carneiro, 15-Jan-2015.)
(𝐴𝑉 → (𝐴 ∈ Fin ↔ (#‘𝐴) ∈ ℕ0))

Theoremnfile 13145 The size of any infinite set is always greater than or equal to the size of any set. (Contributed by AV, 13-Nov-2020.)
((𝐴𝑉𝐵𝑊 ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ≤ (#‘𝐵))

Theoremhashvnfin 13146 A set of finite size is a finite set. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑆𝑉𝑁 ∈ ℕ0) → ((#‘𝑆) = 𝑁𝑆 ∈ Fin))

Theoremhashnfinnn0 13147 The size of an infinite set is not a nonnegative integer. (Contributed by Alexander van der Vekens, 21-Dec-2017.) (Proof shortened by Alexander van der Vekens, 18-Jan-2018.)
((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) ∉ ℕ0)

Theoremisfinite4 13148 A finite set is equinumerous to the range of integers from one up to the hash value of the set. In other words, counting objects with natural numbers works if and only if it is a finite collection. (Contributed by Richard Penner, 26-Feb-2020.)
(𝐴 ∈ Fin ↔ (1...(#‘𝐴)) ≈ 𝐴)

Theoremhasheq0 13149 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.)
(𝐴𝑉 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))

Theoremhashneq0 13150 Two ways of saying a set is not empty. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(𝐴𝑉 → (0 < (#‘𝐴) ↔ 𝐴 ≠ ∅))

Theoremhashgt0n0 13151 If the size of a set is greater than 0, the set is not empty. (Contributed by AV, 5-Aug-2018.) (Prove shortened by AV, 18-Nov-2018.)
((𝐴𝑉 ∧ 0 < (#‘𝐴)) → 𝐴 ≠ ∅)

Theoremhashnncl 13152 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐴 ∈ Fin → ((#‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))

Theoremhash0 13153 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
(#‘∅) = 0

Theoremhashsng 13154 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
(𝐴𝑉 → (#‘{𝐴}) = 1)

Theoremhashen1 13155 A set with only one element is equinumerous to the ordinal number 1. (Contributed by AV, 14-Apr-2019.)
(𝐴𝑉 → ((#‘𝐴) = 1 ↔ 𝐴 ≈ 1𝑜))

Theoremhashrabrsn 13156* The size of a restricted class abstraction restricted to a singleton is a nonnegative integer. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
(#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) ∈ ℕ0

Theoremhashrabsn01 13157* The size of a restricted class abstraction restricted to a singleton is either 0 or 1. (Contributed by Alexander van der Vekens, 3-Sep-2018.)
((#‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1))

Theoremhashrabsn1 13158* 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 13159 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝐹 Fn 𝐴 → (#‘𝐹) = (#‘𝐴))

Theoremfseq1hash 13160 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 13161 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 13162 A short expression for the 𝐺 function of hashgf1o 12765. (Contributed by Mario Carneiro, 24-Jan-2015.)
(# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)

Theoremhashdom 13163 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 13164 Non-strict order relation of the # function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝐴𝐵 → (#‘𝐴) ≤ (#‘𝐵))

Theoremhashsdom 13165 Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) < (#‘𝐵) ↔ 𝐴𝐵))

Theoremhashun 13166 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 13167 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 13168 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 13169 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 13170 The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 13166. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
((𝐴𝑉𝐵𝑊 ∧ (𝐴𝐵) = ∅) → (#‘(𝐴𝐵)) = ((#‘𝐴) +𝑒 (#‘𝐵)))

Theoremhashge0 13171 The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(𝐴𝑉 → 0 ≤ (#‘𝐴))

Theoremhashgt0 13172 The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 0 < (#‘𝐴))

Theoremhashge1 13173 The cardinality of a nonempty set is greater or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)
((𝐴𝑉𝐴 ≠ ∅) → 1 ≤ (#‘𝐴))

Theorem1elfz0hash 13174 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 13175 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 13176 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 13177 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 13178 Obsolete version of hashprg 13177 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 13179 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 13180 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 13181 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 13182 There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 13191, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)
(#‘{0, 1}) = 2

Theoremhashle00 13183 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 13184* 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 13185* 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 13186 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 13187 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘1𝑜) = 1

Theoremhash2 13188 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘2𝑜) = 2

Theoremhash3 13189 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘3𝑜) = 3

Theoremhash4 13190 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
(#‘4𝑜) = 4

Theorempr0hash2ex 13191 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 13192 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 13193 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 13194 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 13195 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 13196 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 13197 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 13198 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 13199 The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.)
((#‘{𝐴}) = 0 ∨ (#‘{𝐴}) = 1)

Theoremhashsnle1 13200 The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.)
(#‘{𝐴}) ≤ 1

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