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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hashresfn 13701 | Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
⊢ (♯ ↾ 𝐴) Fn 𝐴 | ||
Theorem | dmhashres 13702 | Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017.) |
⊢ dom (♯ ↾ 𝐴) = 𝐴 | ||
Theorem | hashnn0pnf 13703 | 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 13700? (Contributed by Alexander van der Vekens, 6-Dec-2017.) |
⊢ (𝑀 ∈ 𝑉 → ((♯‘𝑀) ∈ ℕ0 ∨ (♯‘𝑀) = +∞)) | ||
Theorem | hashnnn0genn0 13704 | 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 13705 | The size of a set is never minus infinity. (Contributed by Alexander van der Vekens, 21-Dec-2017.) |
⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ≠ -∞) | ||
Theorem | hashv01gt1 13706 | 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 13707 | 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 13708 | 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 13709 | 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 13710* | 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 13711* | 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 | focdmex 13712 | The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ V) | ||
Theorem | hasheqf1oi 13713* | 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 13714 | 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 13715 | 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 13716 | 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 13717 | 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 13718 | Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.) |
⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | ||
Theorem | hashxrcl 13719 | Extended real closure of the ♯ function. (Contributed by Mario Carneiro, 22-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ*) | ||
Theorem | hashclb 13720 | Reverse closure of the ♯ function. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (♯‘𝐴) ∈ ℕ0)) | ||
Theorem | nfile 13721 | 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 13722 | 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 13723 | 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 13724 | 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 13725 | Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) |
⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | ||
Theorem | hashneq0 13726 | 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 13727 | 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 13728 | Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) | ||
Theorem | hash0 13729 | The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.) |
⊢ (♯‘∅) = 0 | ||
Theorem | hashelne0d 13730 | 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 13731 | 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 13732 | 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 13733 | 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 13734* | 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 13735* | 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 13736* | 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 13737 | A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) |
⊢ (𝐹 Fn 𝐴 → (♯‘𝐹) = (♯‘𝐴)) | ||
Theorem | fseq1hash 13738 | 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 13739 | 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 13740 | A short expression for the 𝐺 function of hashgf1o 13340. (Contributed by Mario Carneiro, 24-Jan-2015.) |
⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | ||
Theorem | hashdom 13741 | 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 13742 | 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 13743 | Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) < (♯‘𝐵) ↔ 𝐴 ≺ 𝐵)) | ||
Theorem | hashun 13744 | 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 13745 | 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 13746 | 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 13747 | 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 13748 | The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 13744. (Contributed by Alexander van der Vekens, 21-Dec-2017.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +𝑒 (♯‘𝐵))) | ||
Theorem | hashge0 13749 | 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 13750 | The cardinality of a nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 0 < (♯‘𝐴)) | ||
Theorem | hashge1 13751 | 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 13752 | 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 13753 | 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 13754 | 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 13755 | 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 13754. (Contributed by BTernaryTau, 9-Sep-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ 𝐴 → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) +𝑒 1))) | ||
Theorem | hashunsnggt 13756 | 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 13757 | 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 13758 | 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 13759 | 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 13760 | 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 13761 | There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 13770, 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 13762 | 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 13763* | 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 13764* | 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 13765 | 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 13766 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ (♯‘1o) = 1 | ||
Theorem | hash2 13767 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ (♯‘2o) = 2 | ||
Theorem | hash3 13768 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ (♯‘3o) = 3 | ||
Theorem | hash4 13769 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ (♯‘4o) = 4 | ||
Theorem | pr0hash2ex 13770 | 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 13771 | 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 13772 | 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 13773 | 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 13774 | 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 13775 | 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 13776 | 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 13777 | 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 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2)) | ||
Theorem | hashsn01 13778 | The size of a singleton is either 0 or 1. (Contributed by AV, 23-Feb-2021.) |
⊢ ((♯‘{𝐴}) = 0 ∨ (♯‘{𝐴}) = 1) | ||
Theorem | hashsnle1 13779 | The size of a singleton is less than or equal to 1. (Contributed by AV, 23-Feb-2021.) |
⊢ (♯‘{𝐴}) ≤ 1 | ||
Theorem | hashsnlei 13780 | 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 13781* | 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 13782* | 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 13783 | 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 13784* | 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 13785* | 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 13786* | 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 13787 | 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 13788 | 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 13789 | 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 13790 | 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 13791 | 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 13792 | 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 13793 | 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 13794 | 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 13795 | Lemma for hashxp 13796. (Contributed by Paul Chapman, 30-Nov-2012.) |
⊢ 𝐵 ∈ Fin ⇒ ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))) | ||
Theorem | hashxp 13796 | 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 13797 | 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 13798 | 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 13799 | 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 13800 | 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)) |
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