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Definition | df-suc 4301 | Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1". Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Our definition is a generalization to classes. Although it is not conventional to use it with proper classes, it has no effect on a proper class (sucprc 4342). Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134 and the definition of successor in [Mendelson] p. 246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal"). The definition of successor of [Enderton] p. 68 denotes the operation with a plus-sign superscript. (Contributed by NM, 30-Aug-1993.) |
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Theorem | ordeq 4302 | Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.) |
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Theorem | elong 4303 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
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Theorem | elon 4304 | An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
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Theorem | eloni 4305 | An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.) |
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Theorem | elon2 4306 | An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.) |
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Theorem | limeq 4307 | Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
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Theorem | ordtr 4308 | An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.) |
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Theorem | ordelss 4309 | An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.) |
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Theorem | trssord 4310 | A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.) |
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Theorem | ordelord 4311 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.) |
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Theorem | tron 4312 | The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.) |
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Theorem | ordelon 4313 | An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.) |
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Theorem | onelon 4314 | An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.) |
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Theorem | ordin 4315 | The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.) |
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Theorem | onin 4316 | The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.) |
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Theorem | onelss 4317 | An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
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Theorem | ordtr1 4318 | Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) |
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Theorem | ontr1 4319 | Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.) |
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Theorem | onintss 4320* | If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.) |
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Theorem | ord0 4321 | The empty set is an ordinal class. (Contributed by NM, 11-May-1994.) |
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Theorem | 0elon 4322 | The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.) |
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Theorem | inton 4323 | The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.) |
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Theorem | nlim0 4324 | The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
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Theorem | limord 4325 | A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.) |
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Theorem | limuni 4326 | A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.) |
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Theorem | limuni2 4327 | The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.) |
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Theorem | 0ellim 4328 | A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.) |
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Theorem | limelon 4329 | A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.) |
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Theorem | onn0 4330 | The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.) |
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Theorem | onm 4331 | The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.) |
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Theorem | suceq 4332 | Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
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Theorem | elsuci 4333 |
Membership in a successor. This one-way implication does not require that
either ![]() ![]() |
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Theorem | elsucg 4334 | Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.) |
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Theorem | elsuc2g 4335 |
Variant of membership in a successor, requiring that ![]() ![]() |
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Theorem | elsuc 4336 | Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.) |
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Theorem | elsuc2 4337 | Membership in a successor. (Contributed by NM, 15-Sep-2003.) |
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Theorem | nfsuc 4338 | Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.) |
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Theorem | elelsuc 4339 | Membership in a successor. (Contributed by NM, 20-Jun-1998.) |
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Theorem | sucel 4340* | Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.) |
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Theorem | suc0 4341 | The successor of the empty set. (Contributed by NM, 1-Feb-2005.) |
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Theorem | sucprc 4342 | A proper class is its own successor. (Contributed by NM, 3-Apr-1995.) |
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Theorem | unisuc 4343 | A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.) |
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Theorem | unisucg 4344 | A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by Jim Kingdon, 18-Aug-2019.) |
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Theorem | sssucid 4345 | A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.) |
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Theorem | sucidg 4346 | Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
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Theorem | sucid 4347 | A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
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Theorem | nsuceq0g 4348 | No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.) |
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Theorem | eqelsuc 4349 | A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.) |
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Theorem | iunsuc 4350* | Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
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Theorem | suctr 4351 | The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) |
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Theorem | trsuc 4352 | A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
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Theorem | trsucss 4353 | A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.) |
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Theorem | sucssel 4354 | A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.) |
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Theorem | orduniss 4355 | An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.) |
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Theorem | onordi 4356 | An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.) |
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Theorem | ontrci 4357 | An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) |
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Theorem | oneli 4358 | A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
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Theorem | onelssi 4359 | A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.) |
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Theorem | onelini 4360 | An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.) |
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Theorem | oneluni 4361 | An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
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Theorem | onunisuci 4362 | An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) |
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Axiom | ax-un 4363* |
Axiom of Union. An axiom of Intuitionistic Zermelo-Fraenkel set theory.
It states that a set ![]() ![]() ![]() This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4057), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 264). The union of a class df-uni 3745 should not be confused with the union of two classes df-un 3080. Their relationship is shown in unipr 3758. (Contributed by NM, 23-Dec-1993.) |
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Theorem | zfun 4364* | Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) |
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Theorem | axun2 4365* |
A variant of the Axiom of Union ax-un 4363. For any set ![]() ![]() ![]() ![]() |
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Theorem | uniex2 4366* |
The Axiom of Union using the standard abbreviation for union. Given any
set ![]() ![]() |
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Theorem | uniex 4367 |
The Axiom of Union in class notation. This says that if ![]() ![]() ![]() ![]() ![]() |
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Theorem | vuniex 4368 | The union of a setvar is a set. (Contributed by BJ, 3-May-2021.) |
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Theorem | uniexg 4369 |
The ZF Axiom of Union in class notation, in the form of a theorem
instead of an inference. We use the antecedent ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | unex 4370 | The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.) |
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Theorem | unexb 4371 | Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.) |
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Theorem | unexg 4372 | A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) |
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Theorem | tpexg 4373 | An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.) |
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Theorem | unisn3 4374* | Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.) |
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Theorem | abnexg 4375* |
Sufficient condition for a class abstraction to be a proper class. The
class ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | abnex 4376* | Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 4377 and pwnex 4378. See the comment of abnexg 4375. (Contributed by BJ, 2-May-2021.) |
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Theorem | snnex 4377* | The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) |
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Theorem | pwnex 4378* | The class of all power sets is a proper class. See also snnex 4377. (Contributed by BJ, 2-May-2021.) |
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Theorem | opeluu 4379 | Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.) |
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Theorem | uniuni 4380* | Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.) |
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Theorem | eusv1 4381* |
Two ways to express single-valuedness of a class expression
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Theorem | eusvnf 4382* |
Even if ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eusvnfb 4383* |
Two ways to say that ![]() ![]() ![]() ![]() ![]() |
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Theorem | eusv2i 4384* |
Two ways to express single-valuedness of a class expression
![]() ![]() ![]() ![]() |
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Theorem | eusv2nf 4385* |
Two ways to express single-valuedness of a class expression
![]() ![]() ![]() ![]() |
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Theorem | eusv2 4386* |
Two ways to express single-valuedness of a class expression
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Theorem | reusv1 4387* |
Two ways to express single-valuedness of a class expression
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Theorem | reusv3i 4388* | Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.) |
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Theorem | reusv3 4389* |
Two ways to express single-valuedness of a class expression
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Theorem | alxfr 4390* |
Transfer universal quantification from a variable ![]() ![]() ![]() |
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Theorem | ralxfrd 4391* |
Transfer universal quantification from a variable ![]() ![]() ![]() |
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Theorem | rexxfrd 4392* |
Transfer universal quantification from a variable ![]() ![]() ![]() |
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Theorem | ralxfr2d 4393* |
Transfer universal quantification from a variable ![]() ![]() ![]() |
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Theorem | rexxfr2d 4394* |
Transfer universal quantification from a variable ![]() ![]() ![]() |
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Theorem | ralxfr 4395* |
Transfer universal quantification from a variable ![]() ![]() ![]() |
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Theorem | ralxfrALT 4396* |
Transfer universal quantification from a variable ![]() ![]() ![]() |
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Theorem | rexxfr 4397* |
Transfer existence from a variable ![]() ![]() ![]() |
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Theorem | rabxfrd 4398* |
Class builder membership after substituting an expression ![]() ![]() ![]() ![]() |
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Theorem | rabxfr 4399* |
Class builder membership after substituting an expression ![]() ![]() ![]() ![]() |
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Theorem | reuhypd 4400* | A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.) |
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