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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | exmidsssnc 4201* |
Excluded middle in terms of subsets of a singleton. This is similar to
exmid01 4196 but lets you choose any set as the element of
the singleton
rather than just ![]() ![]() |
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Theorem | exmid0el 4202 |
Excluded middle is equivalent to decidability of ![]() |
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Theorem | exmidel 4203* | 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 4204* | 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 6917 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 4205* | Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4204 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.) |
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Theorem | exmid1stab 4206* |
If every proposition is stable, excluded middle follows. We are
thinking of ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Axiom | ax-pr 4207* | 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 4122). (Contributed by NM, 14-Nov-2006.) |
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Theorem | zfpair2 4208 | Derive the abbreviated version of the Axiom of Pairing from ax-pr 4207. (Contributed by NM, 14-Nov-2006.) |
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Theorem | prexg 4209 | 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 3702, prprc1 3700, and prprc2 3701. (Contributed by Jim Kingdon, 16-Sep-2018.) |
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Theorem | snelpwi 4210 | 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 4211 | 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 4212 | 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 4213* | 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 4214 | 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 4215 | 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 4216 | Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) |
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Theorem | pwtr 4217 | 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 4218* | An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.) |
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Theorem | ssext 4219* | 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 4220* | Negation of subclass relationship. Compare nssr 3215. (Contributed by Jim Kingdon, 17-Sep-2018.) |
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Theorem | pweqb 4221 | 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 4222* | 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 4223 | The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.) |
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Theorem | mss 4224* | An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.) |
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Theorem | exss 4225* | 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 4226 | An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.) |
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Theorem | opex 4227 | 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 4228 | An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.) |
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Theorem | elop 4229 | 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 4230 | 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 4231 | 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 4232* | An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.) |
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Theorem | opnzi 4233 | An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4232). (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opth1 4234 | 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 4235 |
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 4236 |
Ordered pair theorem. ![]() ![]() |
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Theorem | opthg2 4237 | Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opth2 4238 | Ordered pair theorem. (Contributed by NM, 21-Sep-2014.) |
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Theorem | otth2 4239 | 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 4240 | Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | eqvinop 4241* | 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 4242* |
Substitution of class ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | copsex2t 4243* | Closed theorem form of copsex2g 4244. (Contributed by NM, 17-Feb-2013.) |
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Theorem | copsex2g 4244* | Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) |
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Theorem | copsex4g 4245* | An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.) |
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Theorem | 0nelop 4246 | A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opeqex 4247 | Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.) |
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Theorem | opcom 4248 | An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.) |
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Theorem | moop2 4249* | "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 4250 | Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) |
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Theorem | opeqpr 4251 | Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) |
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Theorem | euotd 4252* | Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.) |
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Theorem | uniop 4253 | 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 4254 | 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 4255 | 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 4256* | Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.) |
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Theorem | opelopabsbALT 4257* | The law of concretion in terms of substitutions. Less general than opelopabsb 4258, 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 4258* | 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 4259* | The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) |
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Theorem | opelopabt 4260* | Closed theorem form of opelopab 4269. (Contributed by NM, 19-Feb-2013.) |
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Theorem | opelopabga 4261* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | brabga 4262* | The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | opelopab2a 4263* | Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | opelopaba 4264* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | braba 4265* | The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.) |
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Theorem | opelopabg 4266* | 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 4267* | 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 4268* | 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 4269* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.) |
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Theorem | brab 4270* | The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) |
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Theorem | opelopabaf 4271* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4269 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 4272* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4269 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008.) |
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Theorem | ssopab2 4273 | 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 4274 | 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 4275 | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.) |
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Theorem | ssopab2dv 4276* | 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 4277 | Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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Theorem | opabm 4278* | Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.) |
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Theorem | iunopab 4279* | Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
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Theorem | pwin 4280 | 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 4281 | 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 4282 | 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 4283 | 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 4284 | 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 4285 | Extend class notation to include the epsilon relation. |
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Syntax | cid 4286 | Extend the definition of a class to include identity relation. |
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Definition | df-eprel 4287* |
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 4288 | The epsilon relation and membership are the same. General version of epel 4290. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
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Theorem | epelc 4289 | The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.) |
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Theorem | epel 4290 | The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.) |
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Definition | df-id 4291* |
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 4631), but here we introduce a few
related notions we will use to develop ordinals. The class variable | ||
Syntax | wpo 4292 |
Extend wff notation to include the strict partial ordering predicate.
Read: ' ![]() ![]() |
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Syntax | wor 4293 |
Extend wff notation to include the strict linear ordering predicate.
Read: ' ![]() ![]() |
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Definition | df-po 4294* |
Define the strict partial order predicate. Definition of [Enderton]
p. 168. The expression ![]() ![]() ![]() ![]() ![]() |
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Definition | df-iso 4295* |
Define the strict linear order predicate. The expression ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | poss 4296 | 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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Theorem | poeq1 4297 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
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Theorem | poeq2 4298 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
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Theorem | nfpo 4299 | Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
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Theorem | nfso 4300 | Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
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