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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hashnnn0genn0 14301 | 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 14302 | The size of a set is never minus infinity. (Contributed by Alexander van der Vekens, 21-Dec-2017.) |
| ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ≠ -∞) | ||
| Theorem | hashv01gt1 14303 | 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 14304 | 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 14305 | 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 14306 | 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 14307* | 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 14308* | 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 14309* | 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 14310 | 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 14311 | 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 14312 | 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 14313 | 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 14314 | Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.) |
| ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | ||
| Theorem | hashxrcl 14315 | Extended real closure of the ♯ function. (Contributed by Mario Carneiro, 22-Apr-2015.) |
| ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ*) | ||
| Theorem | hashclb 14316 | Reverse closure of the ♯ function. (Contributed by Mario Carneiro, 15-Jan-2015.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (♯‘𝐴) ∈ ℕ0)) | ||
| Theorem | nfile 14317 | 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 14318 | 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 14319 | 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 14320 | 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 14321 | 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 14322 | 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 14323 | 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 14324 | Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.) |
| ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) | ||
| Theorem | hash0 14325 | The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.) |
| ⊢ (♯‘∅) = 0 | ||
| Theorem | hashelne0d 14326 | 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 14327 | 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 14328 | 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 14329 | 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 14330* | 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 14331* | 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 14332* | 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 14333 | A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) |
| ⊢ (𝐹 Fn 𝐴 → (♯‘𝐹) = (♯‘𝐴)) | ||
| Theorem | fseq1hash 14334 | 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 14335 | 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 14336 | A short expression for the 𝐺 function of hashgf1o 13934. (Contributed by Mario Carneiro, 24-Jan-2015.) |
| ⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | ||
| Theorem | hashdom 14337 | 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 14338 | 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 14339 | Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) < (♯‘𝐵) ↔ 𝐴 ≺ 𝐵)) | ||
| Theorem | hashun 14340 | 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 14341 | 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 14342 | 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 14343 | 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 14344 | The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 14340. (Contributed by Alexander van der Vekens, 21-Dec-2017.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +𝑒 (♯‘𝐵))) | ||
| Theorem | hashge0 14345 | 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 14346 | The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 0 < (♯‘𝐴)) | ||
| Theorem | hashge1 14347 | 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 14348 | 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 14349 | 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 14350 | 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 14351 | 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 14350. (Contributed by BTernaryTau, 9-Sep-2023.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ 𝐴 → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) +𝑒 1))) | ||
| Theorem | hashunsnggt 14352 | 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 14353 | 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 14354 | 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 14355 | 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 14356 | 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 14357 | There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 14366, 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 14358 | 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 14359* | 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 14360* | 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 14361 | 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 14362 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
| ⊢ (♯‘1o) = 1 | ||
| Theorem | hash2 14363 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
| ⊢ (♯‘2o) = 2 | ||
| Theorem | hash3 14364 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
| ⊢ (♯‘3o) = 3 | ||
| Theorem | hash4 14365 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
| ⊢ (♯‘4o) = 4 | ||
| Theorem | pr0hash2ex 14366 | 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 14367 | 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 14368 | 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 14369 | 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 14370 | 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 14371 | 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 14372 | 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 14373 | 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 14374 | The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.) |
| ⊢ ((♯‘{𝐴}) = 0 ∨ (♯‘{𝐴}) = 1) | ||
| Theorem | hashsnle1 14375 | The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.) |
| ⊢ (♯‘{𝐴}) ≤ 1 | ||
| Theorem | hashsnlei 14376 | 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 14377* | 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 14378* | 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 14379 | 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 14380* | 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 14381* | 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 14382* | 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 14383 | 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 14384 | 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 14385 | 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 14386 | 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 14387 | 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 14388 | 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 14389 | 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 14390 | 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 14391 | Lemma for hashxp 14392. (Contributed by Paul Chapman, 30-Nov-2012.) |
| ⊢ 𝐵 ∈ Fin ⇒ ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))) | ||
| Theorem | hashxp 14392 | 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 14393 | 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 14394 | 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))) | ||
| Theorem | hashfun 14395 | 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))) | ||
| Theorem | hashres 14396 | 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 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (♯‘(𝐴 ↾ 𝐵)) = (♯‘𝐵)) | ||
| Theorem | hashreshashfun 14397 | The number of elements of a finite function expressed by a restriction. (Contributed by AV, 15-Dec-2021.) |
| ⊢ ((Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (♯‘𝐴) = ((♯‘(𝐴 ↾ 𝐵)) + (♯‘(dom 𝐴 ∖ 𝐵)))) | ||
| Theorem | hashimarn 14398 | 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 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹))) | ||
| Theorem | hashimarni 14399 | 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 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ 𝑃 = (𝐸 “ ran 𝐹) ∧ (♯‘𝑃) = 𝑁) → (♯‘𝐹) = 𝑁)) | ||
| Theorem | hashfundm 14400 | The size of a set function is equal to the size of its domain. (Contributed by BTernaryTau, 30-Sep-2023.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (♯‘𝐹) = (♯‘dom 𝐹)) | ||
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