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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | difprsn2 4801 | 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 4802 | 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 4803 | 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 4804 | 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 4805 | 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 4806 | (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4798. (Contributed by David Moews, 1-May-2017.) |
| ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵) | ||
| Theorem | difsnpss 4807 | (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 4808 | 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 4809 | 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 4810 | 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 4811 | 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 4812 | 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 4813 | Compute the power set of the power set of the empty set. (See pw0 4812 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4900, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.) |
| ⊢ 𝒫 {∅} = {∅, {∅}} | ||
| Theorem | snsspr1 4814 | A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.) |
| ⊢ {𝐴} ⊆ {𝐴, 𝐵} | ||
| Theorem | snsspr2 4815 | A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.) |
| ⊢ {𝐵} ⊆ {𝐴, 𝐵} | ||
| Theorem | snsstp1 4816 | A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) |
| ⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶} | ||
| Theorem | snsstp2 4817 | A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) |
| ⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶} | ||
| Theorem | snsstp3 4818 | A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) |
| ⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶} | ||
| Theorem | prssg 4819 | 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 4820 | 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 4821 | 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 4822 | Deduction version of prssi 4821: 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 4823 | 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 4824 | 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 4825 | 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 4826 | The subsets of a singleton. (Contributed by NM, 24-Apr-2004.) |
| ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) | ||
| Theorem | ssunsn2 4827 | 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 4902. (Contributed by Mario Carneiro, 2-Jul-2016.) |
| ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))) | ||
| Theorem | ssunsn 4828 | 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 4829* | 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 | eqsnd 4830* | Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.) (Proof shortened by SN, 3-Jul-2025.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = {𝐵}) | ||
| Theorem | eqsndOLD 4831* | Obsolete version of eqsnd 4830 as of 3-Jul-2025. (Contributed by Thierry Arnoux, 10-May-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = {𝐵}) | ||
| Theorem | issn 4832* | 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 4833* | 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 4834 | 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 4835 | 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 4836 | 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 4837 | 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 4838 | 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 4839 | Closed form of sneqr 4840. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.) |
| ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) | ||
| Theorem | sneqr 4840 | 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 4841 | 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 4842* | There exists at most one set whose singleton is equal to a given class. See also moeq 3713. (Contributed by BJ, 24-Sep-2022.) |
| ⊢ ∃*𝑥{𝑥} = 𝐴 | ||
| Theorem | sneqbg 4843 | 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 4844 | The singleton of a class is a subset of its power class. (Contributed by NM, 21-Jun-1993.) |
| ⊢ {𝐴} ⊆ 𝒫 𝐴 | ||
| Theorem | prsspw 4845 | 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 4846 | 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 4847 | 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 4848 | 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 4849 | 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 4850 | 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 4851 | An unordered pair has the ordered pair property (compare opth 5481) 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 4852 | Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 4848. (Contributed by AV, 29-Jan-2021.) (Revised by AV, 18-Sep-2021.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)) | ||
| Theorem | preq12bg 4853 | Closed form of preq12b 4850. (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 4854 | 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 | prneimg2 4855 | Two pairs are not equal if their counterparts are not equal. (Contributed by AV, 5-Sep-2025.) |
| ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ∧ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶)))) | ||
| Theorem | prnebg 4856 | 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 4857 | 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 4858 | 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 4859 | 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 4860 | 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 4861 | 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 4862 | 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 4863 | 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 4864 | 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 4865 | An unordered pair has the ordered pair property (compare opth 5481) under certain conditions. Variant of opthpr 4851 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 4866* | Lemma for elpreqpr 4867. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.) |
| ⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}) | ||
| Theorem | elpreqpr 4867* | Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.) |
| ⊢ (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) | ||
| Theorem | elpreqprb 4868* | 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 4869* | 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 4870 | Rewrite df-op 4633 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 4871 | 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 4872 | 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 4873 | 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 4874 | 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 4875 | Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.) |
| ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | ||
| Theorem | opeq1i 4876 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
| ⊢ 𝐴 = 𝐵 ⇒ ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉 | ||
| Theorem | opeq2i 4877 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
| ⊢ 𝐴 = 𝐵 ⇒ ⊢ 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉 | ||
| Theorem | opeq12i 4878 | 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 4879 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | ||
| Theorem | opeq2d 4880 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) | ||
| Theorem | opeq12d 4881 | 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 4882 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
| ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) | ||
| Theorem | oteq2 4883 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
| ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) | ||
| Theorem | oteq3 4884 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
| ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) | ||
| Theorem | oteq1d 4885 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) | ||
| Theorem | oteq2d 4886 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) | ||
| Theorem | oteq3d 4887 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) | ||
| Theorem | oteq123d 4888 | 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 4889 | Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.) |
| ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 | ||
| Theorem | nfopd 4890 | Deduction version of bound-variable hypothesis builder nfop 4889. This shows how the deduction version of a not-free theorem such as nfop 4889 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 4891 | 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 4892 | The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. Closed form of opid 4893. (Contributed by Peter Mazsa, 22-Jul-2019.) (Avoid depending on this detail.) |
| ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}}) | ||
| Theorem | opid 4893 | The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. Inference form of opidg 4892. (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 4894* | 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))) | ||
| Theorem | 2ralunsn 4895* | Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) |
| ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))) | ||
| Theorem | opprc 4896 | Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
| ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | ||
| Theorem | opprc1 4897 | Expansion of an ordered pair when the first member is a proper class. See also opprc 4896. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
| ⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝐵〉 = ∅) | ||
| Theorem | opprc2 4898 | Expansion of an ordered pair when the second member is a proper class. See also opprc 4896. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.) |
| ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) | ||
| Theorem | oprcl 4899 | If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) |
| ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
| Theorem | pwsn 4900 | The power set of a singleton. (Contributed by NM, 5-Jun-2006.) |
| ⊢ 𝒫 {𝐴} = {∅, {𝐴}} | ||
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