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Type | Label | Description |
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Statement | ||
Theorem | reg3exmidlemwe 4501* |
Lemma for reg3exmid 4502. Our counterexample ![]() ![]() |
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Theorem | reg3exmid 4502* |
If any inhabited set satisfying df-wetr 4264 for ![]() |
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Theorem | dcextest 4503* |
If it is decidable whether ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | tfi 4504* |
The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring]
p. 39. This principle states that if ![]() ![]() ![]() ![]() (Contributed by NM, 18-Feb-2004.) |
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Theorem | tfis 4505* |
Transfinite Induction Schema. If all ordinal numbers less than a given
number ![]() |
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Theorem | tfis2f 4506* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
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Theorem | tfis2 4507* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
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Theorem | tfis3 4508* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
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Theorem | tfisi 4509* | A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.) |
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Axiom | ax-iinf 4510* | Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.) |
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Theorem | zfinf2 4511* | A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.) |
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Syntax | com 4512 | Extend class notation to include the class of natural numbers. |
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Definition | df-iom 4513* |
Define the class of natural numbers as the smallest inductive set, which
is valid provided we assume the Axiom of Infinity. Definition 6.3 of
[Eisenberg] p. 82.
Note: the natural numbers We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7067. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4514 instead for naming consistency with set.mm. (New usage is discouraged.) |
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Theorem | dfom3 4514* | Alias for df-iom 4513. Use it instead of df-iom 4513 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
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Theorem | omex 4515 | The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
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Theorem | peano1 4516 | Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.) |
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Theorem | peano2 4517 | The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) |
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Theorem | peano3 4518 | The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) |
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Theorem | peano4 4519 | Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.) |
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Theorem | peano5 4520* | The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as theorem findes 4525. (Contributed by NM, 18-Feb-2004.) |
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Theorem | find 4521* |
The Principle of Finite Induction (mathematical induction). Corollary
7.31 of [TakeutiZaring] p. 43.
The simpler hypothesis shown here was
suggested in an email from "Colin" on 1-Oct-2001. The
hypothesis states
that ![]() ![]() ![]() ![]() ![]() |
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Theorem | finds 4522* | Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.) |
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Theorem | finds2 4523* | Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.) |
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Theorem | finds1 4524* | Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.) |
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Theorem | findes 4525 | Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.) |
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Theorem | nn0suc 4526* | A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.) |
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Theorem | elnn 4527 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
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Theorem | ordom 4528 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
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Theorem | omelon2 4529 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
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Theorem | omelon 4530 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
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Theorem | nnon 4531 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
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Theorem | nnoni 4532 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
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Theorem | nnord 4533 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
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Theorem | omsson 4534 |
Omega is a subset of ![]() |
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Theorem | limom 4535 | Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.) |
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Theorem | peano2b 4536 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
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Theorem | nnsuc 4537* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
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Theorem | nnsucpred 4538 | The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.) |
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Theorem | nndceq0 4539 | A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.) |
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Theorem | 0elnn 4540 | A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.) |
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Theorem | nn0eln0 4541 | A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.) |
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Theorem | nnregexmid 4542* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4458 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6403 or nntri3or 6397), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.) |
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Theorem | omsinds 4543* |
Strong (or "total") induction principle over ![]() |
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Theorem | nnpredcl 4544 |
The predecessor of a natural number is a natural number. This theorem
is most interesting when the natural number is a successor (as seen in
theorems like onsucuni2 4487) but also holds when it is ![]() |
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Syntax | cxp 4545 | Extend the definition of a class to include the cross product. |
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Syntax | ccnv 4546 | Extend the definition of a class to include the converse of a class. |
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Syntax | cdm 4547 | Extend the definition of a class to include the domain of a class. |
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Syntax | crn 4548 | Extend the definition of a class to include the range of a class. |
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Syntax | cres 4549 |
Extend the definition of a class to include the restriction of a class.
(Read: The restriction of ![]() ![]() |
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Syntax | cima 4550 |
Extend the definition of a class to include the image of a class. (Read:
The image of ![]() ![]() |
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Syntax | ccom 4551 |
Extend the definition of a class to include the composition of two
classes. (Read: The composition of ![]() ![]() |
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Syntax | wrel 4552 |
Extend the definition of a wff to include the relation predicate. (Read:
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Definition | df-xp 4553* |
Define the cross product of two classes. Definition 9.11 of [Quine]
p. 64. For example, ( { 1 , 5 } ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-rel 4554 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4997 and dfrel3 5004. (Contributed by NM, 1-Aug-1994.) |
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Definition | df-cnv 4555* |
Define the converse of a class. Definition 9.12 of [Quine] p. 64. The
converse of a binary relation swaps its arguments, i.e., if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-co 4556* |
Define the composition of two classes. Definition 6.6(3) of
[TakeutiZaring] p. 24. Note that
Definition 7 of [Suppes] p. 63
reverses ![]() ![]() ![]() |
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Definition | df-dm 4557* |
Define the domain of a class. Definition 3 of [Suppes] p. 59. For
example, F = { ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-rn 4558 |
Define the range of a class. For example, F = { ![]() ![]() ![]() ![]() ![]() |
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Definition | df-res 4559 |
Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring]
p. 24. For example ( F = { ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-ima 4560 |
Define the image of a class (as restricted by another class).
Definition 6.6(2) of [TakeutiZaring] p. 24. For example, ( F = {
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Theorem | xpeq1 4561 | Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
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Theorem | xpeq2 4562 | Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) |
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Theorem | elxpi 4563* | Membership in a cross product. Uses fewer axioms than elxp 4564. (Contributed by NM, 4-Jul-1994.) |
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Theorem | elxp 4564* | Membership in a cross product. (Contributed by NM, 4-Jul-1994.) |
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Theorem | elxp2 4565* | Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
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Theorem | xpeq12 4566 | Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
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Theorem | xpeq1i 4567 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
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Theorem | xpeq2i 4568 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
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Theorem | xpeq12i 4569 | Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
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Theorem | xpeq1d 4570 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
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Theorem | xpeq2d 4571 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
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Theorem | xpeq12d 4572 | Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.) |
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Theorem | sqxpeqd 4573 | Equality deduction for a Cartesian square, see Wikipedia "Cartesian product", https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power. (Contributed by AV, 13-Jan-2020.) |
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Theorem | nfxp 4574 | Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | 0nelxp 4575 | The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | 0nelelxp 4576 | A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.) |
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Theorem | opelxp 4577 | Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | brxp 4578 | Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.) |
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Theorem | opelxpi 4579 | Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.) |
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Theorem | opelxpd 4580 | Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | opelxp1 4581 | The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opelxp2 4582 | The second member of an ordered pair of classes in a cross product belongs to second cross product argument. (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | otelxp1 4583 | The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) |
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Theorem | rabxp 4584* | Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.) |
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Theorem | brrelex12 4585 | A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | brrelex1 4586 | A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | brrelex 4587 | A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | brrelex2 4588 | A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | brrelex12i 4589 | Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.) |
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Theorem | brrelex1i 4590 | The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.) |
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Theorem | brrelex2i 4591 | The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | nprrel 4592 | No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.) |
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Theorem | 0nelrel 4593 | A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) |
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Theorem | fconstmpt 4594* |
Representation of a constant function using the mapping operation.
(Note that ![]() ![]() |
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Theorem | vtoclr 4595* | Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opelvvg 4596 | Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.) |
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Theorem | opelvv 4597 | Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opthprc 4598 | Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.) |
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Theorem | brel 4599 | Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | brab2a 4600* | Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.) |
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