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Type | Label | Description |
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Statement | ||
Theorem | pwntru 4201 | A slight strengthening of pwtrufal 14832. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.) |
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Theorem | exmid1dc 4202* |
A convenience theorem for proving that something implies EXMID.
Think of this as an alternative to using a proposition, as in proofs
like undifexmid 4195 or ordtriexmid 4522. In this context ![]() ![]() ![]() ![]() ![]() |
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Theorem | exmidn0m 4203* | Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.) |
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Theorem | exmidsssn 4204* | Excluded middle is equivalent to the biconditionalized version of sssnr 3755 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.) |
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Theorem | exmidsssnc 4205* |
Excluded middle in terms of subsets of a singleton. This is similar to
exmid01 4200 but lets you choose any set as the element of
the singleton
rather than just ![]() ![]() |
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Theorem | exmid0el 4206 |
Excluded middle is equivalent to decidability of ![]() |
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Theorem | exmidel 4207* | Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.) |
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Theorem | exmidundif 4208* | 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 3505 and undifdcss 6924 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.) |
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Theorem | exmidundifim 4209* | Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4208 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.) |
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Theorem | exmid1stab 4210* |
If every proposition is stable, excluded middle follows. We are
thinking of ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Axiom | ax-pr 4211* | 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 4126). (Contributed by NM, 14-Nov-2006.) |
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Theorem | zfpair2 4212 | Derive the abbreviated version of the Axiom of Pairing from ax-pr 4211. (Contributed by NM, 14-Nov-2006.) |
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Theorem | prexg 4213 | 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 3704, prprc1 3702, and prprc2 3703. (Contributed by Jim Kingdon, 16-Sep-2018.) |
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Theorem | snelpwi 4214 | A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.) |
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Theorem | snelpw 4215 | A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.) |
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Theorem | prelpwi 4216 | A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.) |
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Theorem | rext 4217* | A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.) |
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Theorem | sspwb 4218 | Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) |
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Theorem | unipw 4219 | 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.) |
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Theorem | pwel 4220 | Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) |
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Theorem | pwtr 4221 | A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.) |
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Theorem | ssextss 4222* | An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.) |
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Theorem | ssext 4223* | 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.) |
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Theorem | nssssr 4224* | Negation of subclass relationship. Compare nssr 3217. (Contributed by Jim Kingdon, 17-Sep-2018.) |
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Theorem | pweqb 4225 | Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) |
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Theorem | intid 4226* | The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) |
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Theorem | euabex 4227 | The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.) |
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Theorem | mss 4228* | An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.) |
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Theorem | exss 4229* | Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.) |
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Theorem | opexg 4230 | An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.) |
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Theorem | opex 4231 | An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.) |
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Theorem | otexg 4232 | An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.) |
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Theorem | elop 4233 | 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.) |
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Theorem | opi1 4234 | One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opi2 4235 | One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opm 4236* | An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.) |
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Theorem | opnzi 4237 | An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4236). (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opth1 4238 | Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opth 4239 |
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
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Theorem | opthg 4240 |
Ordered pair theorem. ![]() ![]() |
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Theorem | opthg2 4241 | Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opth2 4242 | Ordered pair theorem. (Contributed by NM, 21-Sep-2014.) |
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Theorem | otth2 4243 | Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | otth 4244 | Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | eqvinop 4245* | A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.) |
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Theorem | copsexg 4246* |
Substitution of class ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | copsex2t 4247* | Closed theorem form of copsex2g 4248. (Contributed by NM, 17-Feb-2013.) |
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Theorem | copsex2g 4248* | Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) |
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Theorem | copsex4g 4249* | An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.) |
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Theorem | 0nelop 4250 | A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opeqex 4251 | Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.) |
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Theorem | opcom 4252 | An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.) |
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Theorem | moop2 4253* | "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opeqsn 4254 | Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) |
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Theorem | opeqpr 4255 | Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) |
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Theorem | euotd 4256* | Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.) |
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Theorem | uniop 4257 | 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.) |
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Theorem | uniopel 4258 | Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opabid 4259 | 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.) |
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Theorem | elopab 4260* | Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.) |
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Theorem | opelopabsbALT 4261* | The law of concretion in terms of substitutions. Less general than opelopabsb 4262, 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.) |
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Theorem | opelopabsb 4262* | The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.) |
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Theorem | brabsb 4263* | The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) |
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Theorem | opelopabt 4264* | Closed theorem form of opelopab 4273. (Contributed by NM, 19-Feb-2013.) |
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Theorem | opelopabga 4265* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | brabga 4266* | The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | opelopab2a 4267* | Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | opelopaba 4268* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | braba 4269* | The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.) |
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Theorem | opelopabg 4270* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.) |
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Theorem | brabg 4271* | The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.) |
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Theorem | opelopab2 4272* | Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.) |
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Theorem | opelopab 4273* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.) |
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Theorem | brab 4274* | The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) |
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Theorem | opelopabaf 4275* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4273 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.) |
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Theorem | opelopabf 4276* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4273 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008.) |
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Theorem | ssopab2 4277 | Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.) |
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Theorem | ssopab2b 4278 | Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
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Theorem | ssopab2i 4279 | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.) |
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Theorem | ssopab2dv 4280* | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
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Theorem | eqopab2b 4281 | Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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Theorem | opabm 4282* | Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.) |
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Theorem | iunopab 4283* | Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
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Theorem | pwin 4284 | 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.) |
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Theorem | pwunss 4285 | 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.) |
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Theorem | pwssunim 4286 | 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.) |
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Theorem | pwundifss 4287 | Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.) |
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Theorem | pwunim 4288 | 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.) |
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Syntax | cep 4289 | Extend class notation to include the epsilon relation. |
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Syntax | cid 4290 | Extend the definition of a class to include identity relation. |
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Definition | df-eprel 4291* |
Define the epsilon relation. Similar to Definition 6.22 of
[TakeutiZaring] p. 30. The
epsilon relation and set membership are the
same, that is, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | epelg 4292 | The epsilon relation and membership are the same. General version of epel 4294. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
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Theorem | epelc 4293 | The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.) |
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Theorem | epel 4294 | The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.) |
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Definition | df-id 4295* |
Define the identity relation. Definition 9.15 of [Quine] p. 64. For
example, 5 ![]() ![]() ![]() |
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We have not yet defined relations (df-rel 4635), but here we introduce a few
related notions we will use to develop ordinals. The class variable | ||
Syntax | wpo 4296 |
Extend wff notation to include the strict partial ordering predicate.
Read: ' ![]() ![]() |
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Syntax | wor 4297 |
Extend wff notation to include the strict linear ordering predicate.
Read: ' ![]() ![]() |
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Definition | df-po 4298* |
Define the strict partial order predicate. Definition of [Enderton]
p. 168. The expression ![]() ![]() ![]() ![]() ![]() |
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Definition | df-iso 4299* |
Define the strict linear order predicate. The expression ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | poss 4300 | Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
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