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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | exmid0el 4201 | Excluded middle is equivalent to decidability of ∅ being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.) |
⊢ (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥) | ||
Theorem | exmidel 4202* | Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.) |
⊢ (EXMID ↔ ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦) | ||
Theorem | exmidundif 4203* | Excluded middle is equivalent to every subset having a complement. That is, the union of a subset and its relative complement being the whole set. Although special cases such as undifss 3503 and undifdcss 6916 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.) |
⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)) | ||
Theorem | exmidundifim 4204* | Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4203 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.) |
⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)) | ||
Theorem | exmid1stab 4205* | If every proposition is stable, excluded middle follows. We are thinking of 𝑥 as a proposition and 𝑥 = {∅} as "𝑥 is true". (Contributed by Jim Kingdon, 28-Nov-2023.) |
⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅}) ⇒ ⊢ (𝜑 → EXMID) | ||
Axiom | ax-pr 4206* | The Axiom of Pairing of IZF set theory. Axiom 2 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4121). (Contributed by NM, 14-Nov-2006.) |
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | ||
Theorem | zfpair2 4207 | Derive the abbreviated version of the Axiom of Pairing from ax-pr 4206. (Contributed by NM, 14-Nov-2006.) |
⊢ {𝑥, 𝑦} ∈ V | ||
Theorem | prexg 4208 | The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3701, prprc1 3699, and prprc2 3700. (Contributed by Jim Kingdon, 16-Sep-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) | ||
Theorem | snelpwi 4209 | A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.) |
⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) | ||
Theorem | snelpw 4210 | A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵) | ||
Theorem | prelpwi 4211 | A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶) | ||
Theorem | rext 4212* | A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.) |
⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦) | ||
Theorem | sspwb 4213 | Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) |
⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) | ||
Theorem | unipw 4214 | A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.) |
⊢ ∪ 𝒫 𝐴 = 𝐴 | ||
Theorem | pwel 4215 | Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) |
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) | ||
Theorem | pwtr 4216 | A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.) |
⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴) | ||
Theorem | ssextss 4217* | An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.) |
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) | ||
Theorem | ssext 4218* | An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.) |
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵)) | ||
Theorem | nssssr 4219* | Negation of subclass relationship. Compare nssr 3215. (Contributed by Jim Kingdon, 17-Sep-2018.) |
⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵) | ||
Theorem | pweqb 4220 | Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) |
⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | intid 4221* | The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴} | ||
Theorem | euabex 4222 | The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.) |
⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) | ||
Theorem | mss 4223* | An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.) |
⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑧 𝑧 ∈ 𝑥)) | ||
Theorem | exss 4224* | Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) | ||
Theorem | opexg 4225 | An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) | ||
Theorem | opex 4226 | An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ 〈𝐴, 𝐵〉 ∈ V | ||
Theorem | otexg 4227 | An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.) |
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) | ||
Theorem | elop 4228 | An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) | ||
Theorem | opi1 4229 | One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 | ||
Theorem | opi2 4230 | One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 | ||
Theorem | opm 4231* | An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.) |
⊢ (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | opnzi 4232 | An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4231). (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ 〈𝐴, 𝐵〉 ≠ ∅ | ||
Theorem | opth1 4233 | Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐴 = 𝐶) | ||
Theorem | opth 4234 | The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | opthg 4235 | Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | opthg2 4236 | Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | opth2 4237 | Ordered pair theorem. (Contributed by NM, 21-Sep-2014.) |
⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | otth2 4238 | Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑅 ∈ V ⇒ ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) | ||
Theorem | otth 4239 | Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑅 ∈ V ⇒ ⊢ (〈𝐴, 𝐵, 𝑅〉 = 〈𝐶, 𝐷, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) | ||
Theorem | eqvinop 4240* | A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 = 〈𝐵, 𝐶〉 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 = 〈𝐵, 𝐶〉)) | ||
Theorem | copsexg 4241* | Substitution of class 𝐴 for ordered pair 〈𝑥, 𝑦〉. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | ||
Theorem | copsex2t 4242* | Closed theorem form of copsex2g 4243. (Contributed by NM, 17-Feb-2013.) |
⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | copsex2g 4243* | Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | copsex4g 4244* | An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.) |
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤((〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 〈𝐶, 𝐷〉 = 〈𝑧, 𝑤〉) ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | 0nelop 4245 | A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ¬ ∅ ∈ 〈𝐴, 𝐵〉 | ||
Theorem | opeqex 4246 | Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.) |
⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))) | ||
Theorem | opcom 4247 | An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ 𝐴 = 𝐵) | ||
Theorem | moop2 4248* | "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 | ||
Theorem | opeqsn 4249 | Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) | ||
Theorem | opeqpr 4250 | Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) | ||
Theorem | euotd 4251* | Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → (𝜓 ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶))) ⇒ ⊢ (𝜑 → ∃!𝑥∃𝑎∃𝑏∃𝑐(𝑥 = 〈𝑎, 𝑏, 𝑐〉 ∧ 𝜓)) | ||
Theorem | uniop 4252 | The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} | ||
Theorem | uniopel 4253 | Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → ∪ 〈𝐴, 𝐵〉 ∈ ∪ 𝐶) | ||
Theorem | opabid 4254 | The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | ||
Theorem | elopab 4255* | Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.) |
⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | ||
Theorem | opelopabsbALT 4256* | The law of concretion in terms of substitutions. Less general than opelopabsb 4257, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) | ||
Theorem | opelopabsb 4257* | The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.) |
⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | ||
Theorem | brabsb 4258* | The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) |
⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | ||
Theorem | opelopabt 4259* | Closed theorem form of opelopab 4268. (Contributed by NM, 19-Feb-2013.) |
⊢ ((∀𝑥∀𝑦(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥∀𝑦(𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | opelopabga 4260* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓)) | ||
Theorem | brabga 4261* | The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓)) | ||
Theorem | opelopab2a 4262* | Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜓)) | ||
Theorem | opelopaba 4263* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) | ||
Theorem | braba 4264* | The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝜓) | ||
Theorem | opelopabg 4265* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | brabg 4266* | The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) | ||
Theorem | opelopab2 4267* | Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜒)) | ||
Theorem | opelopab 4268* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) | ||
Theorem | brab 4269* | The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝜒) | ||
Theorem | opelopabaf 4270* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4268 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) | ||
Theorem | opelopabf 4271* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4268 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜒 & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) | ||
Theorem | ssopab2 4272 | Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) | ||
Theorem | ssopab2b 4273 | Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | ssopab2i 4274 | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} | ||
Theorem | ssopab2dv 4275* | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜒}) | ||
Theorem | eqopab2b 4276 | Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) | ||
Theorem | opabm 4277* | Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.) |
⊢ (∃𝑧 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦𝜑) | ||
Theorem | iunopab 4278* | Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} | ||
Theorem | pwin 4279 | The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) |
⊢ 𝒫 (𝐴 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵) | ||
Theorem | pwunss 4280 | The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) |
⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) | ||
Theorem | pwssunim 4281 | The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵)) | ||
Theorem | pwundifss 4282 | Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.) |
⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) ⊆ 𝒫 (𝐴 ∪ 𝐵) | ||
Theorem | pwunim 4283 | The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by Jim Kingdon, 30-Sep-2018.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝒫 (𝐴 ∪ 𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵)) | ||
Syntax | cep 4284 | Extend class notation to include the epsilon relation. |
class E | ||
Syntax | cid 4285 | Extend the definition of a class to include identity relation. |
class I | ||
Definition | df-eprel 4286* | Define the epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is, (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) when 𝐵 is a set by epelg 4287. Thus, 5 E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.) |
⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | ||
Theorem | epelg 4287 | The epsilon relation and membership are the same. General version of epel 4289. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | epelc 4288 | The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) | ||
Theorem | epel 4289 | The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.) |
⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | ||
Definition | df-id 4290* | Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5 I 5 and ¬ 4 I 5. (Contributed by NM, 13-Aug-1995.) |
⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | ||
We have not yet defined relations (df-rel 4630), but here we introduce a few related notions we will use to develop ordinals. The class variable 𝑅 is no different from other class variables, but it reminds us that typically it represents what we will later call a "relation". | ||
Syntax | wpo 4291 | Extend wff notation to include the strict partial ordering predicate. Read: ' 𝑅 is a partial order on 𝐴.' |
wff 𝑅 Po 𝐴 | ||
Syntax | wor 4292 | Extend wff notation to include the strict linear ordering predicate. Read: ' 𝑅 orders 𝐴.' |
wff 𝑅 Or 𝐴 | ||
Definition | df-po 4293* | Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression 𝑅 Po 𝐴 means 𝑅 is a partial order on 𝐴. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | ||
Definition | df-iso 4294* | Define the strict linear order predicate. The expression 𝑅 Or 𝐴 is true if relationship 𝑅 orders 𝐴. The property 𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) is called weak linearity by Proposition 11.2.3 of [HoTT], p. (varies). If we assumed excluded middle, it would be equivalent to trichotomy, 𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥. (Contributed by NM, 21-Jan-1996.) (Revised by Jim Kingdon, 4-Oct-2018.) |
⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))) | ||
Theorem | poss 4295 | Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Po 𝐵 → 𝑅 Po 𝐴)) | ||
Theorem | poeq1 4296 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴)) | ||
Theorem | poeq2 4297 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵)) | ||
Theorem | nfpo 4298 | Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Po 𝐴 | ||
Theorem | nfso 4299 | Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Or 𝐴 | ||
Theorem | pocl 4300 | Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))) |
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