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Theorem List for Metamath Proof Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem0nelop 4401 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremopeqex 4402 Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)

Theoremoteqex2 4403 Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.)

Theoremoteqex 4404 Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopcom 4405 An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)

Theoremmoop2 4406* "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeqsn 4407 Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Theoremopeqpr 4408 Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)

Theoremmosubopt 4409* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)

Theoremmosubop 4410* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)

Theoremeuop2 4411* Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.)

Theoremeuotd 4412* Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)

Theoremopthwiener 4413 Justification theorem for the ordered pair definition in Norbert Wiener, "A simplification of the logic of relations," Proc. of the Cambridge Philos. Soc., 1914, vol. 17, pp.387-390. It is also shown as a definition in [Enderton] p. 36 and as Exercise 4.8(b) of [Mendelson] p. 230. It is meaningful only for classes that exist as sets (i.e. are not proper classes). See df-op 3780 for other ordered pair definitions. (Contributed by NM, 28-Sep-2003.)

Theoremuniop 4414 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.)

Theoremuniopel 4415 Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

2.3.4  Ordered-pair class abstractions (cont.)

Theoremopabid 4416 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.)

Theoremelopab 4417* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)

TheoremopelopabsbOLD 4418* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

TheorembrabsbOLD 4419* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremopelopabsb 4420* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)

Theorembrabsb 4421* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)

Theoremopelopabt 4422* Closed theorem form of opelopab 4431. (Contributed by NM, 19-Feb-2013.)

Theoremopelopabga 4423* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theorembrabga 4424* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremopelopab2a 4425* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremopelopaba 4426* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theorembraba 4427* The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)

Theoremopelopabg 4428* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theorembrabg 4429* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremopelopab2 4430* Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremopelopab 4431* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)

Theorembrab 4432* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)

Theoremopelopabaf 4433* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4431 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.)

Theoremopelopabf 4434* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4431 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.)

Theoremssopab2 4435 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)

Theoremssopab2b 4436 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theoremssopab2i 4437 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)

Theoremssopab2dv 4438* Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremeqopab2b 4439 Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremopabn0 4440 Non-empty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.)

Theoremiunopab 4441* Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)

2.3.5  Power class of union and intersection

Theorempwin 4442 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.)

Theorempwunss 4443 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.)

Theorempwssun 4444 The power class of the union of two classes is a subset of the union of their power classes, iff one class is a subclass of the other. Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)

Theorempwundif 4445 Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.)

Theorempwun 4446 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 NM, 23-Nov-2003.)

2.3.6  Epsilon and identity relations

Syntaxcep 4447 Extend class notation to include the epsilon relation.

Syntaxcid 4448 Extend the definition of a class to include identity relation.

Definitiondf-eprel 4449* Define the epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is, when is a set by epelg 4450. Thus, (ex-eprel 21603). (Contributed by NM, 13-Aug-1995.)

Theoremepelg 4450 The epsilon relation and membership are the same. General version of epel 4452. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremepelc 4451 The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremepel 4452 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)

Definitiondf-id 4453* Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, and (ex-id 21604). (Contributed by NM, 13-Aug-1995.)

Theoremdfid3 4454 A stronger version of df-id 4453 that doesn't require and to be distinct. Ordinarily, we wouldn't use this as a definition, since non-distinct dummy variables would make soundness verification more difficult (as the proof here shows). The proof can be instructive in showing how distinct variable requirements may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.)

Theoremdfid2 4455 Alternate definition of the identity relation. (Contributed by NM, 15-Mar-2007.)

2.3.7  Partial and complete ordering

Syntaxwpo 4456 Extend wff notation to include the strict partial ordering predicate. Read: ' is a partial order on .'

Syntaxwor 4457 Extend wff notation to include the strict complete ordering predicate. Read: ' orders .'

Definitiondf-po 4458* Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression means is a partial order on . For example, is true, while is false (ex-po 21605). (Contributed by NM, 16-Mar-1997.)

Definitiondf-so 4459* Define the strict complete (linear) order predicate. The expression is true if relationship orders . For example, is true (ltso 9103). Equivalent to Definition 6.19(1) of [TakeutiZaring] p. 29. (Contributed by NM, 21-Jan-1996.)

Theoremposs 4460 Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Theorempoeq1 4461 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)

Theorempoeq2 4462 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)

Theoremnfpo 4463 Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremnfso 4464 Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theorempocl 4465 Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)

Theoremispod 4466* Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)

Theoremswopolem 4467* Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremswopo 4468* A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theorempoirr 4469 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)

Theorempotr 4470 A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)

Theorempo2nr 4471 A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)

Theorempo3nr 4472 A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)

Theorempo0 4473 Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorempofun 4474* A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)

Theoremsopo 4475 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)

Theoremsoss 4476 Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremsoeq1 4477 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)

Theoremsoeq2 4478 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)

Theoremsonr 4479 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)

Theoremsotr 4480 A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)

Theoremsolin 4481 A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.)

Theoremso2nr 4482 A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)

Theoremso3nr 4483 A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)

Theoremsotric 4484 A strict order relation satisfies strict trichotomy. (Contributed by NM, 19-Feb-1996.)

Theoremsotrieq 4485 Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremsotrieq2 4486 Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.)

Theoremsotr2 4487 A transitivity relation. (Read and implies .) (Contributed by Mario Carneiro, 10-May-2013.)

Theoremissod 4488* An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremissoi 4489* An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremisso2i 4490* Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremso0 4491 Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremsomo 4492* A totally ordered set has at most one minimal element. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by NM, 16-Jun-2017.)

2.3.8  Founded and well-ordering relations

Syntaxwfr 4493 Extend wff notation to include the well-founded predicate. Read: ' is a well-founded relation on .'

Syntaxwse 4494 Extend wff notation to include the set-like predicate. Read: ' is set-like on .'
Se

Syntaxwwe 4495 Extend wff notation to include the well-ordering predicate. Read: ' well-orders .'

Definitiondf-fr 4496* Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 4502 and dffr3 5190. (Contributed by NM, 3-Apr-1994.)

Definitiondf-se 4497* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
Se

Definitiondf-we 4498 Define the well-ordering predicate. For an alternate definition, see dfwe2 4716. (Contributed by NM, 3-Apr-1994.)

Theoremfri 4499* Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)

Theoremseex 4500* The -preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)
Se

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