P. 144 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14301-14400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hashfxnn0 14301	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*
Theorem	hashf 14302	The size function maps all finite sets to their cardinality, as members of ℕ0, and infinite sets to +∞. TODO-AV: mark as OBSOLETE and replace it by hashfxnn0 14301? (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.) (Proof shortened by AV, 24-Oct-2021.)
⊢ ♯:V⟶(ℕ0 ∪ {+∞})
Theorem	hashxnn0 14303	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.)
⊢ (𝑀 ∈ 𝑉 → (♯‘𝑀) ∈ ℕ0*)
Theorem	hashresfn 14304	Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ (♯ ↾ 𝐴) Fn 𝐴
Theorem	dmhashres 14305	Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017.)
⊢ dom (♯ ↾ 𝐴) = 𝐴
Theorem	hashnn0pnf 14306	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 14303? (Contributed by Alexander van der Vekens, 6-Dec-2017.)
⊢ (𝑀 ∈ 𝑉 → ((♯‘𝑀) ∈ ℕ0 ∨ (♯‘𝑀) = +∞))
Theorem	hashnnn0genn0 14307	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.)
⊢ ((𝑀 ∈ 𝑉 ∧ (♯‘𝑀) ∉ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (♯‘𝑀))
Theorem	hashnemnf 14308	The size of a set is never minus infinity. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ≠ -∞)
Theorem	hashv01gt1 14309	The size of a set is either 0 or 1 or greater than 1. (Contributed by Alexander van der Vekens, 29-Dec-2017.)
⊢ (𝑀 ∈ 𝑉 → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)))
Theorem	hashfz1 14310	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)
Theorem	hashen 14311	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) ↔ 𝐴 ≈ 𝐵))
Theorem	hasheni 14312	Equinumerous sets have the same number of elements (even if they are not finite). (Contributed by Mario Carneiro, 15-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 → (♯‘𝐴) = (♯‘𝐵))
Theorem	hasheqf1o 14313*	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:𝐴–1-1-onto→𝐵))
Theorem	fiinfnf1o 14314*	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:𝐴–1-1-onto→𝐵)
Theorem	hasheqf1oi 14315*	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.)
⊢ (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵)))
Theorem	hashf1rn 14316	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))
Theorem	hasheqf1od 14317	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)    ⇒   ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵))
Theorem	fz1eqb 14318	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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((1...𝑀) = (1...𝑁) ↔ 𝑀 = 𝑁))
Theorem	hashcard 14319	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))
Theorem	hashcl 14320	Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
Theorem	hashxrcl 14321	Extended real closure of the ♯ function. (Contributed by Mario Carneiro, 22-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ*)
Theorem	hashclb 14322	Reverse closure of the ♯ function. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (♯‘𝐴) ∈ ℕ0))
Theorem	nfile 14323	The size of any infinite set is always greater than or equal to the size of any set. (Contributed by AV, 13-Nov-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ (♯‘𝐵))
Theorem	hashvnfin 14324	A set of finite size is a finite set. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑆) = 𝑁 → 𝑆 ∈ Fin))
Theorem	hashnfinnn0 14325	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) ∉ ℕ0)
Theorem	isfinite4 14326	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)
Theorem	hasheq0 14327	Two ways of saying a set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
Theorem	hashneq0 14328	Two ways of saying a set is not empty. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
⊢ (𝐴 ∈ 𝑉 → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅))
Theorem	hashgt0n0 14329	If the size of a set is greater than 0, the set is not empty. (Contributed by AV, 5-Aug-2018.) (Proof shortened by AV, 18-Nov-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 0 < (♯‘𝐴)) → 𝐴 ≠ ∅)
Theorem	hashnncl 14330	Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))
Theorem	hash0 14331	The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
⊢ (♯‘∅) = 0
Theorem	hashelne0d 14332	A set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ¬ (♯‘𝐴) = 0)
Theorem	hashsng 14333	The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1)
Theorem	hashen1 14334	A set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o))
Theorem	hash1elsn 14335	A set of size 1 with a known element is the singleton of that element. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → (♯‘𝐴) = 1)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 = {𝐵})
Theorem	hashrabrsn 14336*	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
Theorem	hashrabsn01 14337*	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))
Theorem	hashrabsn1 14338*	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)
Theorem	hashfn 14339	A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ (𝐹 Fn 𝐴 → (♯‘𝐹) = (♯‘𝐴))
Theorem	fseq1hash 14340	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 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
Theorem	hashgadd 14341	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)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))
Theorem	hashgval2 14342	A short expression for the 𝐺 function of hashgf1o 13940. (Contributed by Mario Carneiro, 24-Jan-2015.)
⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
Theorem	hashdom 14343	Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
Theorem	hashdomi 14344	Non-strict order relation of the ♯ function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ (𝐴 ≼ 𝐵 → (♯‘𝐴) ≤ (♯‘𝐵))
Theorem	hashsdom 14345	Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) < (♯‘𝐵) ↔ 𝐴 ≺ 𝐵))
Theorem	hashun 14346	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 ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
Theorem	hashun2 14347	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) → (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵)))
Theorem	hashun3 14348	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) → (♯‘(𝐴 ∪ 𝐵)) = (((♯‘𝐴) + (♯‘𝐵)) − (♯‘(𝐴 ∩ 𝐵))))
Theorem	hashinfxadd 14349	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞)
Theorem	hashunx 14350	The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 14346. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +𝑒 (♯‘𝐵)))
Theorem	hashge0 14351	The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
⊢ (𝐴 ∈ 𝑉 → 0 ≤ (♯‘𝐴))
Theorem	hashgt0 14352	The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 0 < (♯‘𝐴))
Theorem	hashge1 14353	The cardinality of a nonempty set is greater than or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 1 ≤ (♯‘𝐴))
Theorem	1elfz0hash 14354	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)))
Theorem	hashnn0n0nn 14355	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.)
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) ∧ ((♯‘𝑉) = 𝑌 ∧ 𝑁 ∈ 𝑉)) → 𝑌 ∈ ℕ)
Theorem	hashunsng 14356	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.)
⊢ (𝐵 ∈ 𝑉 → ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) + 1)))
Theorem	hashunsngx 14357	The size of the union of a set with a disjoint singleton is the extended real addition of the size of the set and 1, analogous to hashunsng 14356. (Contributed by BTernaryTau, 9-Sep-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ 𝐴 → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) +𝑒 1)))
Theorem	hashunsnggt 14358	The size of a set is greater than a nonnegative integer N if and only if the size of the union of that set with a disjoint singleton is greater than N + 1. (Contributed by BTernaryTau, 10-Sep-2023.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑁 ∈ ℕ0) ∧ ¬ 𝐵 ∈ 𝐴) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) < (♯‘(𝐴 ∪ {𝐵}))))
Theorem	hashprg 14359	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
Theorem	elprchashprn2 14360	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)
Theorem	hashprb 14361	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 ∧ 𝑀 ≠ 𝑁) ↔ (♯‘{𝑀, 𝑁}) = 2)
Theorem	hashprdifel 14362	The elements of an unordered pair of size 2 are different sets. (Contributed by AV, 27-Jan-2020.)
⊢ 𝑆 = {𝐴, 𝐵}    ⇒   ⊢ ((♯‘𝑆) = 2 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵))
Theorem	prhash2ex 14363	There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 14372, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)
⊢ (♯‘{0, 1}) = 2
Theorem	hashle00 14364	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.)
⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) ≤ 0 ↔ 𝑉 = ∅))
Theorem	hashgt0elex 14365*	If the size of a set is greater than zero, then the set must contain at least one element. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
⊢ ((𝑉 ∈ 𝑊 ∧ 0 < (♯‘𝑉)) → ∃𝑥 𝑥 ∈ 𝑉)
Theorem	hashgt0elexb 14366*	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.)
⊢ (𝑉 ∈ 𝑊 → (0 < (♯‘𝑉) ↔ ∃𝑥 𝑥 ∈ 𝑉))
Theorem	hashp1i 14367	Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 = suc 𝐴    &   ⊢ (♯‘𝐴) = 𝑀    &   ⊢ (𝑀 + 1) = 𝑁    ⇒   ⊢ (♯‘𝐵) = 𝑁
Theorem	hash1 14368	Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (♯‘1o) = 1
Theorem	hash2 14369	Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (♯‘2o) = 2
Theorem	hash3 14370	Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (♯‘3o) = 3
Theorem	hash4 14371	Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (♯‘4o) = 4
Theorem	pr0hash2ex 14372	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
Theorem	hashss 14373	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → (♯‘𝐵) ≤ (♯‘𝐴))
Theorem	prsshashgt1 14374	The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 ∈ 𝑈) → ({𝐴, 𝐵} ⊆ 𝐶 → 2 ≤ (♯‘𝐶)))
Theorem	hashin 14375	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.)
⊢ (𝐴 ∈ 𝑉 → (♯‘(𝐴 ∩ 𝐵)) ≤ (♯‘𝐴))
Theorem	hashssdif 14376	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 ∧ 𝐵 ⊆ 𝐴) → (♯‘(𝐴 ∖ 𝐵)) = ((♯‘𝐴) − (♯‘𝐵)))
Theorem	hashdif 14377	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 → (♯‘(𝐴 ∖ 𝐵)) = ((♯‘𝐴) − (♯‘(𝐴 ∩ 𝐵))))
Theorem	hashdifsn 14378	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 ∧ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1))
Theorem	hashdifpr 14379	The size of the difference of a finite set and a proper pair of its elements is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)
⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2))
Theorem	hashsn01 14380	The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.)
⊢ ((♯‘{𝐴}) = 0 ∨ (♯‘{𝐴}) = 1)
Theorem	hashsnle1 14381	The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.)
⊢ (♯‘{𝐴}) ≤ 1
Theorem	hashsnlei 14382	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)
Theorem	hash1snb 14383*	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.)
⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎}))
Theorem	euhash1 14384*	The size of a set is 1 in terms of existential uniqueness. (Contributed by Alexander van der Vekens, 8-Feb-2018.)
⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 ↔ ∃!𝑎 𝑎 ∈ 𝑉))
Theorem	hash1n0 14385	If the size of a set is 1 the set is not empty. (Contributed by AV, 23-Dec-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = 1) → 𝐴 ≠ ∅)
Theorem	hashgt12el 14386*	In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
⊢ ((𝑉 ∈ 𝑊 ∧ 1 < (♯‘𝑉)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑎 ≠ 𝑏)
Theorem	hashgt12el2 14387*	In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
⊢ ((𝑉 ∈ 𝑊 ∧ 1 < (♯‘𝑉) ∧ 𝐴 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 𝐴 ≠ 𝑏)
Theorem	hashgt23el 14388*	A set with more than two elements has at least three different elements. (Contributed by BTernaryTau, 21-Sep-2023.)
⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
Theorem	hashunlei 14389	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 = (𝐴 ∪ 𝐵)    &   ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝐾)    &   ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑀)    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ (𝐾 + 𝑀) = 𝑁    ⇒   ⊢ (𝐶 ∈ Fin ∧ (♯‘𝐶) ≤ 𝑁)
Theorem	hashsslei 14390	Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐵 ⊆ 𝐴    &   ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝑁)    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑁)
Theorem	hashfz 14391	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 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1))
Theorem	fzsdom2 14392	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 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶))
Theorem	hashfzo 14393	Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴))
Theorem	hashfzo0 14394	Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝐵 ∈ ℕ0 → (♯‘(0..^𝐵)) = 𝐵)
Theorem	hashfzp1 14395	Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴))
Theorem	hashfz0 14396	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))
Theorem	hashxplem 14397	Lemma for hashxp 14398. (Contributed by Paul Chapman, 30-Nov-2012.)
⊢ 𝐵 ∈ Fin    ⇒   ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
Theorem	hashxp 14398	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)))
Theorem	hashmap 14399	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.) (Proof shortened by AV, 18-Jul-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ↑m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))
Theorem	hashpw 14400	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)))