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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pwuni 4001 | A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) |
⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | ||
Theorem | undifexmid 4002* | Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3350 and undifdcss 6585 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.) |
⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Syntax | wem 4003 | Formula for an abbreviation of excluded middle. |
wff EXMID | ||
Definition | df-exmid 4004 |
The expression EXMID will be used as a
readable shorthand for any
form of the law of the excluded middle; this is a useful shorthand
largely because it hides statements of the form "for any
proposition" in
a system which can only quantify over sets, not propositions.
To see how this compares with other ways of expressing excluded middle, compare undifexmid 4002 with exmidundif 4009. The former may be more recognizable as excluded middle because it is in terms of propositions, and the proof may be easier to follow for much the same reason (it just has to show 𝜑 and ¬ 𝜑 in the the relevant parts of the proof). The latter, however, has the key advantage of being able to prove both directions of the biconditional. To state that excluded middle implies a proposition is hard to do gracefully without EXMID, because there is no way to write a hypothesis 𝜑 ∨ ¬ 𝜑 for an arbitrary proposition; instead the hypothesis would need to be the particular instance of excluded middle which that proof needs. Or to say it another way, EXMID implies DECID 𝜑 by exmidexmid 4005 but there is no good way to express the converse. This definition and how we use it is easiest to understand (and most appropriate to assign the name "excluded middle" to) if we assume ax-sep 3932, in which case EXMID means that all propositions are decidable (see exmidexmid 4005 and notice that it relies on ax-sep 3932). If we instead work with ax-bdsep 11213, EXMID as defined here means that all bounded propositions are decidable. (Contributed by Mario Carneiro and Jim Kingdon, 18-Jun-2022.) |
⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) | ||
Theorem | exmidexmid 4005 |
EXMID implies that an arbitrary proposition is decidable. That is,
EXMID captures the usual meaning of excluded middle when stated in terms
of propositions.
To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 787, peircedc 856, or condc 785. (Contributed by Jim Kingdon, 18-Jun-2022.) |
⊢ (EXMID → DECID 𝜑) | ||
Theorem | exmid01 4006 | Excluded middle is equivalent to saying any subset of {∅} is either ∅ or {∅}. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.) |
⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))) | ||
Theorem | exmid0el 4007 | 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 4008* | 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 4009* | 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 3350 and undifdcss 6585 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)) | ||
Axiom | ax-pr 4010* | 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 3935). (Contributed by NM, 14-Nov-2006.) |
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | ||
Theorem | zfpair2 4011 | Derive the abbreviated version of the Axiom of Pairing from ax-pr 4010. (Contributed by NM, 14-Nov-2006.) |
⊢ {𝑥, 𝑦} ∈ V | ||
Theorem | prexg 4012 | 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 3535, prprc1 3533, and prprc2 3534. (Contributed by Jim Kingdon, 16-Sep-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) | ||
Theorem | snelpwi 4013 | 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 4014 | 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 4015 | 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 4016* | 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 4017 | 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 4018 | 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 4019 | 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 4020 | 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 4021* | 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 4022* | 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 4023* | Negation of subclass relationship. Compare nssr 3073. (Contributed by Jim Kingdon, 17-Sep-2018.) |
⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵) | ||
Theorem | pweqb 4024 | 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 4025* | 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 4026 | The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.) |
⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) | ||
Theorem | mss 4027* | 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 4028* | 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 4029 | An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ V) | ||
Theorem | opex 4030 | 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 4031 | 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 4032 | 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 4033 | 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 4034 | 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 4035* | 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 4036 | An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4035). (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅ | ||
Theorem | opth1 4037 | 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 4038 | 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 4039 | 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 4040 | 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 4041 | Ordered pair theorem. (Contributed by NM, 21-Sep-2014.) |
⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | otth2 4042 | 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 4043 | 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 4044* | 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 4045* | 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 4046* | Closed theorem form of copsex2g 4047. (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 4047* | 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 4048* | 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 4049 | A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ¬ ∅ ∈ ⟨𝐴, 𝐵⟩ | ||
Theorem | opeqex 4050 | 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 4051 | 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 4052* | "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 4053 | 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 4054 | 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 4055* | 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 4056 | 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 4057 | 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 4058 | 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 4059* | Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.) |
⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) | ||
Theorem | opelopabsbALT 4060* | The law of concretion in terms of substitutions. Less general than opelopabsb 4061, 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 4061* | 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 4062* | 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 4063* | Closed theorem form of opelopab 4072. (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 4064* | 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 4065* | 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 4066* | 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 4067* | 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 4068* | 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 4069* | 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 4070* | 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 4071* | 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 4072* | 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 4073* | 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 4074* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4072 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 4075* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4072 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 4076 | 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 4077 | 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 4078 | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} | ||
Theorem | ssopab2dv 4079* | 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 4080 | 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 4081* | Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.) |
⊢ (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦𝜑) | ||
Theorem | iunopab 4082* | 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 4083 | 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 4084 | 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 4085 | 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 4086 | 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 4087 | 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 4088 | Extend class notation to include the epsilon relation. |
class E | ||
Syntax | cid 4089 | Extend the definition of a class to include identity relation. |
class I | ||
Definition | df-eprel 4090* | 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 4091. Thus, 5 E { 1 , 5 }. (Contributed by NM, 13-Aug-1995.) |
⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} | ||
Theorem | epelg 4091 | The epsilon relation and membership are the same. General version of epel 4093. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | epelc 4092 | 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 4093 | The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.) |
⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | ||
Definition | df-id 4094* | 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} | ||
Syntax | wpo 4095 | Extend wff notation to include the strict partial ordering predicate. Read: ' 𝑅 is a partial order on 𝐴.' |
wff 𝑅 Po 𝐴 | ||
Syntax | wor 4096 | Extend wff notation to include the strict linear ordering predicate. Read: ' 𝑅 orders 𝐴.' |
wff 𝑅 Or 𝐴 | ||
Definition | df-po 4097* | 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 4098* | 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 4099 | 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 4100 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴)) |
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