![]() |
Metamath
Proof Explorer Theorem List (p. 49 of 486) | < 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: | ![]() (1-30851) |
![]() (30852-32374) |
![]() (32375-48553) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | neldifsn 4801 | The class 𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.) |
⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}) | ||
Theorem | neldifsnd 4802 | The class 𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) | ||
Theorem | rexdifsn 4803 | Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.) |
⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝜑)) | ||
Theorem | raldifsni 4804 | Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
⊢ (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) | ||
Theorem | raldifsnb 4805 | Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑) | ||
Theorem | eldifvsn 4806 | A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴 ≠ 𝐵)) | ||
Theorem | difsn 4807 | An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵) | ||
Theorem | difprsnss 4808 | Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵} | ||
Theorem | difprsn1 4809 | Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) | ||
Theorem | difprsn2 4810 | Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.) |
⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴}) | ||
Theorem | diftpsn3 4811 | Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵}) | ||
Theorem | difpr 4812 | Removing two elements as pair of elements corresponds to removing each of the two elements as singletons. (Contributed by Alexander van der Vekens, 13-Jul-2018.) |
⊢ (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶}) | ||
Theorem | tpprceq3 4813 | An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.) |
⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | ||
Theorem | tppreqb 4814 | An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.) |
⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵) ↔ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | ||
Theorem | difsnb 4815 | (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4807. (Contributed by David Moews, 1-May-2017.) |
⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) | ||
Theorem | difsnpss 4816 | (𝐵 ∖ {𝐴}) is a proper subclass of 𝐵 if and only if 𝐴 is a member of 𝐵. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵) | ||
Theorem | snssi 4817 | The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.) |
⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | ||
Theorem | snssd 4818 | The singleton of an element of a class is a subset of the class (deduction form). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → {𝐴} ⊆ 𝐵) | ||
Theorem | difsnid 4819 | If we remove a single element from a class then put it back in, we end up with the original class. (Contributed by NM, 2-Oct-2006.) |
⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) | ||
Theorem | eldifeldifsn 4820 | An element of a difference set is an element of the difference with a singleton. (Contributed by AV, 2-Jan-2022.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋})) | ||
Theorem | pw0 4821 | Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ 𝒫 ∅ = {∅} | ||
Theorem | pwpw0 4822 | Compute the power set of the power set of the empty set. (See pw0 4821 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4906, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.) |
⊢ 𝒫 {∅} = {∅, {∅}} | ||
Theorem | snsspr1 4823 | A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.) |
⊢ {𝐴} ⊆ {𝐴, 𝐵} | ||
Theorem | snsspr2 4824 | A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.) |
⊢ {𝐵} ⊆ {𝐴, 𝐵} | ||
Theorem | snsstp1 4825 | A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) |
⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶} | ||
Theorem | snsstp2 4826 | A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) |
⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶} | ||
Theorem | snsstp3 4827 | A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) |
⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶} | ||
Theorem | prssg 4828 | A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)) | ||
Theorem | prss 4829 | A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶) | ||
Theorem | prssi 4830 | A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ⊆ 𝐶) | ||
Theorem | prssd 4831 | Deduction version of prssi 4830: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶) | ||
Theorem | prsspwg 4832 | An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))) | ||
Theorem | ssprss 4833 | A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))) | ||
Theorem | ssprsseq 4834 | A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷})) | ||
Theorem | sssn 4835 | The subsets of a singleton. (Contributed by NM, 24-Apr-2004.) |
⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) | ||
Theorem | ssunsn2 4836 | The property of being sandwiched between two sets naturally splits under union with a singleton. This is the induction hypothesis for the determination of large powersets such as pwtp 4908. (Contributed by Mario Carneiro, 2-Jul-2016.) |
⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))) | ||
Theorem | ssunsn 4837 | Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.) |
⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶}))) | ||
Theorem | eqsn 4838* | Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) | ||
Theorem | issn 4839* | A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.) |
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}) | ||
Theorem | n0snor2el 4840* | A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.) |
⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧})) | ||
Theorem | ssunpr 4841 | Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.) |
⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷})))) | ||
Theorem | sspr 4842 | The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Mario Carneiro, 2-Jul-2016.) |
⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}))) | ||
Theorem | sstp 4843 | The subsets of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.) |
⊢ (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})))) | ||
Theorem | tpss 4844 | An unordered triple of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷) | ||
Theorem | tpssi 4845 | An unordered triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷) | ||
Theorem | sneqrg 4846 | Closed form of sneqr 4847. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) | ||
Theorem | sneqr 4847 | If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) | ||
Theorem | snsssn 4848 | If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵) | ||
Theorem | mosneq 4849* | There exists at most one set whose singleton is equal to a given class. See also moeq 3701. (Contributed by BJ, 24-Sep-2022.) |
⊢ ∃*𝑥{𝑥} = 𝐴 | ||
Theorem | sneqbg 4850 | Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.) |
⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵)) | ||
Theorem | snsspw 4851 | The singleton of a class is a subset of its power class. (Contributed by NM, 21-Jun-1993.) |
⊢ {𝐴} ⊆ 𝒫 𝐴 | ||
Theorem | prsspw 4852 | An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by OpenAI, 25-Mar-2020.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) | ||
Theorem | preq1b 4853 | Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same second element iff the first elements are equal. (Contributed by AV, 18-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵)) | ||
Theorem | preq2b 4854 | Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same first element iff the second elements are equal. (Contributed by AV, 18-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵)) | ||
Theorem | preqr1 4855 | Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) | ||
Theorem | preqr2 4856 | Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 15-Jul-1993.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵) | ||
Theorem | preq12b 4857 | Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) | ||
Theorem | opthpr 4858 | An unordered pair has the ordered pair property (compare opth 5482) under certain conditions. (Contributed by NM, 27-Mar-2007.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (𝐴 ≠ 𝐷 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | preqr1g 4859 | Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 4855. (Contributed by AV, 29-Jan-2021.) (Revised by AV, 18-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)) | ||
Theorem | preq12bg 4860 | Closed form of preq12b 4857. (Contributed by Scott Fenton, 28-Mar-2014.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))) | ||
Theorem | prneimg 4861 | Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.) |
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})) | ||
Theorem | prnebg 4862 | A (proper) pair is not equal to another (maybe improper) pair if and only if an element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 16-Jan-2018.) |
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ {𝐴, 𝐵} ≠ {𝐶, 𝐷})) | ||
Theorem | pr1eqbg 4863 | A (proper) pair is equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶})) | ||
Theorem | pr1nebg 4864 | A (proper) pair is not equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → (𝐴 ≠ 𝐶 ↔ {𝐴, 𝐵} ≠ {𝐵, 𝐶})) | ||
Theorem | preqsnd 4865 | Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.) (Revised by AV, 13-Jun-2022.) (Revised by AV, 16-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) | ||
Theorem | prnesn 4866 | A proper unordered pair is not a (proper or improper) singleton. (Contributed by AV, 13-Jun-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≠ {𝐶}) | ||
Theorem | prneprprc 4867 | A proper unordered pair is not an improper unordered pair. (Contributed by AV, 13-Jun-2022.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}) | ||
Theorem | preqsn 4868 | Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 12-Jun-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) | ||
Theorem | preq12nebg 4869 | Equality relationship for two proper unordered pairs. (Contributed by AV, 12-Jun-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))) | ||
Theorem | prel12g 4870 | Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.) (Revised by AV, 9-Dec-2018.) (Revised by AV, 12-Jun-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))) | ||
Theorem | opthprneg 4871 | An unordered pair has the ordered pair property (compare opth 5482) under certain conditions. Variant of opthpr 4858 in closed form. (Contributed by AV, 13-Jun-2022.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | elpreqprlem 4872* | Lemma for elpreqpr 4873. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.) |
⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}) | ||
Theorem | elpreqpr 4873* | Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.) |
⊢ (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) | ||
Theorem | elpreqprb 4874* | A set is an element of an unordered pair iff there is another (maybe the same) set which is an element of the unordered pair. (Proposed by BJ, 8-Dec-2020.) (Contributed by AV, 9-Dec-2020.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})) | ||
Theorem | elpr2elpr 4875* | For an element 𝐴 of an unordered pair which is a subset of a given set 𝑉, there is another (maybe the same) element 𝑏 of the given set 𝑉 being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}) | ||
Theorem | dfopif 4876 | Rewrite df-op 4640 using if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) |
⊢ 〈𝐴, 𝐵〉 = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | ||
Theorem | dfopg 4877 | Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | ||
Theorem | dfop 4878 | Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} | ||
Theorem | opeq1 4879 | Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | ||
Theorem | opeq2 4880 | Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) | ||
Theorem | opeq12 4881 | Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.) |
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | ||
Theorem | opeq1i 4882 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉 | ||
Theorem | opeq2i 4883 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉 | ||
Theorem | opeq12i 4884 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉 | ||
Theorem | opeq1d 4885 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | ||
Theorem | opeq2d 4886 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) | ||
Theorem | opeq12d 4887 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) | ||
Theorem | oteq1 4888 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) | ||
Theorem | oteq2 4889 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) | ||
Theorem | oteq3 4890 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) | ||
Theorem | oteq1d 4891 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) | ||
Theorem | oteq2d 4892 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) | ||
Theorem | oteq3d 4893 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) | ||
Theorem | oteq123d 4894 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) & ⊢ (𝜑 → 𝐸 = 𝐹) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) | ||
Theorem | nfop 4895 | Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 | ||
Theorem | nfopd 4896 | Deduction version of bound-variable hypothesis builder nfop 4895. This shows how the deduction version of a not-free theorem such as nfop 4895 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) | ||
Theorem | csbopg 4897 | Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.) (Revised by ML, 25-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌〈𝐶, 𝐷〉 = 〈⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷〉) | ||
Theorem | opidg 4898 | The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. Closed form of opid 4899. (Contributed by Peter Mazsa, 22-Jul-2019.) (Avoid depending on this detail.) |
⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}}) | ||
Theorem | opid 4899 | The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. Inference form of opidg 4898. (Contributed by FL, 28-Dec-2011.) (Proof shortened by AV, 16-Feb-2022.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} | ||
Theorem | ralunsn 4900* | Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |