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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iin0r 4101* | If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
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Theorem | intv 4102 | The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.) |
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Theorem | axpweq 4103* | Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4106 is not used by the proof. (Contributed by NM, 22-Jun-2009.) |
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Theorem | bnd 4104* |
A very strong generalization of the Axiom of Replacement (compare
zfrep6 4053). Its strength lies in the rather profound
fact that
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Theorem | bnd2 4105* |
A variant of the Boundedness Axiom bnd 4104 that picks a subset ![]() ![]() |
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Axiom | ax-pow 4106* |
Axiom of Power Sets. An axiom of Intuitionistic Zermelo-Fraenkel set
theory. It states that a set ![]() ![]() ![]() The variant axpow2 4108 uses explicit subset notation. A version using class notation is pwex 4115. (Contributed by NM, 5-Aug-1993.) |
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Theorem | zfpow 4107* | Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) |
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Theorem | axpow2 4108* | A variant of the Axiom of Power Sets ax-pow 4106 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
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Theorem | axpow3 4109* |
A variant of the Axiom of Power Sets ax-pow 4106. For any set ![]() ![]() ![]() ![]() |
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Theorem | el 4110* | Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
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Theorem | vpwex 4111 | Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4112 from vpwex 4111. (Revised by BJ, 10-Aug-2022.) |
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Theorem | pwexg 4112 | Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.) |
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Theorem | pwexd 4113 | Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
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Theorem | abssexg 4114* | Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
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Theorem | pwex 4115 | Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.) |
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Theorem | snexg 4116 |
A singleton whose element exists is a set. The ![]() ![]() ![]() |
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Theorem | snex 4117 | A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
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Theorem | snexprc 4118 |
A singleton whose element is a proper class is a set. The ![]() ![]() ![]() ![]() |
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Theorem | notnotsnex 4119 | A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.) |
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Theorem | p0ex 4120 | The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.) |
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Theorem | pp0ex 4121 |
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Theorem | ord3ex 4122 | The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.) |
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Theorem | dtruarb 4123* |
At least two sets exist (or in terms of first-order logic, the universe
of discourse has two or more objects). This theorem asserts the
existence of two sets which do not equal each other; compare with
dtruex 4482 in which we are given a set ![]() ![]() |
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Theorem | pwuni 4124 | A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) |
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Theorem | undifexmid 4125* | Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3448 and undifdcss 6819 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.) |
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Syntax | wem 4126 | Formula for an abbreviation of excluded middle. |
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Definition | df-exmid 4127 |
The expression EXMID will be used as a readable shorthand for
any
form of the law of the excluded middle; this is a useful shorthand
largely because it hides statements of the form "for any
proposition" in
a system which can only quantify over sets, not propositions.
To see how this compares with other ways of expressing excluded middle,
compare undifexmid 4125 with exmidundif 4137. The former may be more
recognizable as excluded middle because it is in terms of propositions,
and the proof may be easier to follow for much the same reason (it just
has to show This definition and how we use it is easiest to understand (and most appropriate to assign the name "excluded middle" to) if we assume ax-sep 4054, in which case EXMID means that all propositions are decidable (see exmidexmid 4128 and notice that it relies on ax-sep 4054). If we instead work with ax-bdsep 13253, EXMID as defined here means that all bounded propositions are decidable. (Contributed by Mario Carneiro and Jim Kingdon, 18-Jun-2022.) |
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Theorem | exmidexmid 4128 |
EXMID implies that an arbitrary proposition is decidable. That is,
EXMID captures the usual meaning of excluded middle when stated in terms
of propositions.
To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 829, peircedc 900, or condc 839. (Contributed by Jim Kingdon, 18-Jun-2022.) |
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Theorem | exmid01 4129 |
Excluded middle is equivalent to saying any subset of ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | pwntru 4130 | A slight strengthening of pwtrufal 13365. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.) |
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Theorem | exmid1dc 4131* |
A convenience theorem for proving that something implies EXMID.
Think of this as an alternative to using a proposition, as in proofs
like undifexmid 4125 or ordtriexmid 4445. In this context ![]() ![]() ![]() ![]() ![]() |
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Theorem | exmidn0m 4132* | Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.) |
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Theorem | exmidsssn 4133* | Excluded middle is equivalent to the biconditionalized version of sssnr 3688 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.) |
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Theorem | exmidsssnc 4134* |
Excluded middle in terms of subsets of a singleton. This is similar to
exmid01 4129 but lets you choose any set as the element of
the singleton
rather than just ![]() ![]() |
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Theorem | exmid0el 4135 |
Excluded middle is equivalent to decidability of ![]() |
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Theorem | exmidel 4136* | 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 4137* | 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 3448 and undifdcss 6819 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 4138* | Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4137 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.) |
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Axiom | ax-pr 4139* | 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 4057). (Contributed by NM, 14-Nov-2006.) |
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Theorem | zfpair2 4140 | Derive the abbreviated version of the Axiom of Pairing from ax-pr 4139. (Contributed by NM, 14-Nov-2006.) |
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Theorem | prexg 4141 | 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 3641, prprc1 3639, and prprc2 3640. (Contributed by Jim Kingdon, 16-Sep-2018.) |
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Theorem | snelpwi 4142 | 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 4143 | 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 4144 | 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 4145* | 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 4146 | 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 4147 | 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 4148 | Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) |
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Theorem | pwtr 4149 | 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 4150* | An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.) |
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Theorem | ssext 4151* | 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 4152* | Negation of subclass relationship. Compare nssr 3162. (Contributed by Jim Kingdon, 17-Sep-2018.) |
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Theorem | pweqb 4153 | 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 4154* | 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 4155 | The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.) |
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Theorem | mss 4156* | An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.) |
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Theorem | exss 4157* | 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 4158 | An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.) |
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Theorem | opex 4159 | 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 4160 | An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.) |
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Theorem | elop 4161 | 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 4162 | 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 4163 | 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 4164* | An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.) |
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Theorem | opnzi 4165 | An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4164). (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opth1 4166 | 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 4167 |
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 4168 |
Ordered pair theorem. ![]() ![]() |
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Theorem | opthg2 4169 | Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opth2 4170 | Ordered pair theorem. (Contributed by NM, 21-Sep-2014.) |
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Theorem | otth2 4171 | 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 4172 | Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | eqvinop 4173* | 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 4174* |
Substitution of class ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | copsex2t 4175* | Closed theorem form of copsex2g 4176. (Contributed by NM, 17-Feb-2013.) |
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Theorem | copsex2g 4176* | Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) |
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Theorem | copsex4g 4177* | An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.) |
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Theorem | 0nelop 4178 | A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
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Theorem | opeqex 4179 | Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.) |
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Theorem | opcom 4180 | An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.) |
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Theorem | moop2 4181* | "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 4182 | Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) |
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Theorem | opeqpr 4183 | Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) |
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Theorem | euotd 4184* | Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.) |
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Theorem | uniop 4185 | 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 4186 | 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 4187 | 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 4188* | Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.) |
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Theorem | opelopabsbALT 4189* | The law of concretion in terms of substitutions. Less general than opelopabsb 4190, 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 4190* | 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 4191* | The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) |
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Theorem | opelopabt 4192* | Closed theorem form of opelopab 4201. (Contributed by NM, 19-Feb-2013.) |
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Theorem | opelopabga 4193* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | brabga 4194* | The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | opelopab2a 4195* | Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | opelopaba 4196* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | braba 4197* | The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.) |
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Theorem | opelopabg 4198* | 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 4199* | 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 4200* | 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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