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Type | Label | Description |
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Statement | ||
Theorem | finds2 4601* | 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 4602* | 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 4603 | 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 4604* | 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 | elomssom 4605 | A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4606. (Revised by BJ, 7-Aug-2024.) |
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Theorem | elnn 4606 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
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Theorem | ordom 4607 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
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Theorem | omelon2 4608 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
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Theorem | omelon 4609 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
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Theorem | nnon 4610 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
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Theorem | nnoni 4611 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
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Theorem | nnord 4612 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
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Theorem | omsson 4613 |
Omega is a subset of ![]() |
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Theorem | limom 4614 | 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 4615 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
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Theorem | nnsuc 4616* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
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Theorem | nnsucpred 4617 | The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.) |
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Theorem | nndceq0 4618 | 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 4619 | 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 4620 | 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 4621* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4535 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6500 or nntri3or 6494), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.) |
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Theorem | omsinds 4622* |
Strong (or "total") induction principle over ![]() |
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Theorem | nnpredcl 4623 |
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 4564) but also holds when it is ![]() |
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Theorem | nnpredlt 4624 | The predecessor (see nnpredcl 4623) of a nonzero natural number is less than (see df-iord 4367) that number. (Contributed by Jim Kingdon, 14-Sep-2024.) |
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Syntax | cxp 4625 | Extend the definition of a class to include the cross product. |
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Syntax | ccnv 4626 | Extend the definition of a class to include the converse of a class. |
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Syntax | cdm 4627 | Extend the definition of a class to include the domain of a class. |
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Syntax | crn 4628 | Extend the definition of a class to include the range of a class. |
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Syntax | cres 4629 |
Extend the definition of a class to include the restriction of a class.
(Read: The restriction of ![]() ![]() |
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Syntax | cima 4630 |
Extend the definition of a class to include the image of a class. (Read:
The image of ![]() ![]() |
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Syntax | ccom 4631 |
Extend the definition of a class to include the composition of two
classes. (Read: The composition of ![]() ![]() |
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Syntax | wrel 4632 |
Extend the definition of a wff to include the relation predicate. (Read:
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Definition | df-xp 4633* |
Define the Cartesian product of two classes. This is also sometimes
called the "cross product" but that term also has other
meanings; we
intentionally choose a less ambiguous term. Definition 9.11 of [Quine]
p. 64. For example, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-rel 4634 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5080 and dfrel3 5087. (Contributed by NM, 1-Aug-1994.) |
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Definition | df-cnv 4635* |
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. "Converse" is Quine's terminology. Some authors use a "minus one" exponent and call it "inverse", especially when the argument is a function, although this is not in general a genuine inverse. (Contributed by NM, 4-Jul-1994.) |
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Definition | df-co 4636* |
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 4637* |
Define the domain of a class. Definition 3 of [Suppes] p. 59. For
example, F = { ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-rn 4638 |
Define the range of a class. For example, F = { ![]() ![]() ![]() ![]() ![]() |
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Definition | df-res 4639 |
Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring]
p. 24. For example,
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Definition | df-ima 4640 |
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 4641 | Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
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Theorem | xpeq2 4642 | Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) |
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Theorem | elxpi 4643* | Membership in a cross product. Uses fewer axioms than elxp 4644. (Contributed by NM, 4-Jul-1994.) |
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Theorem | elxp 4644* | Membership in a cross product. (Contributed by NM, 4-Jul-1994.) |
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Theorem | elxp2 4645* | Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
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Theorem | xpeq12 4646 | Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
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Theorem | xpeq1i 4647 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
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Theorem | xpeq2i 4648 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
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Theorem | xpeq12i 4649 | Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
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Theorem | xpeq1d 4650 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
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Theorem | xpeq2d 4651 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
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Theorem | xpeq12d 4652 | Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.) |
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Theorem | sqxpeqd 4653 | 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 4654 | 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 4655 | 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 4656 | 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 4657 | 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 4658 | Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.) |
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Theorem | opelxpi 4659 | Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.) |
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Theorem | opelxpd 4660 | Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | opelxp1 4661 | 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 4662 | 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 4663 | 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 4664* | Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.) |
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Theorem | brrelex12 4665 | 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 4666 | 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 4667 | 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 4668 | 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 4669 | 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 4670 | 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 4671 | 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 4672 | No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.) |
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Theorem | 0nelrel 4673 | A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) |
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Theorem | fconstmpt 4674* |
Representation of a constant function using the mapping operation.
(Note that ![]() ![]() |
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Theorem | vtoclr 4675* | 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 4676 | Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.) |
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Theorem | opelvv 4677 | 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 4678 | 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 4679 | 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 4680* | Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.) |
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Theorem | elxp3 4681* | Membership in a cross product. (Contributed by NM, 5-Mar-1995.) |
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Theorem | opeliunxp 4682 | Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.) |
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Theorem | xpundi 4683 | Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.) |
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Theorem | xpundir 4684 | Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.) |
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Theorem | xpiundi 4685* | Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
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Theorem | xpiundir 4686* | Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
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Theorem | iunxpconst 4687* | Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.) |
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Theorem | xpun 4688 | The cross product of two unions. (Contributed by NM, 12-Aug-2004.) |
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Theorem | elvv 4689* | Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.) |
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Theorem | elvvv 4690* | Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.) |
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Theorem | elvvuni 4691 | An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.) |
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Theorem | mosubopt 4692* | "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.) |
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Theorem | mosubop 4693* | "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.) |
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Theorem | brinxp2 4694 | Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | brinxp 4695 | Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.) |
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Theorem | poinxp 4696 | Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.) |
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Theorem | soinxp 4697 | Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.) |
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Theorem | seinxp 4698 | Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.) |
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Theorem | posng 4699 | Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.) |
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Theorem | sosng 4700 | Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.) |
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